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EXPERIMENTAL  PSYCHOLOGY 


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WORKS  BY 
EDWARD  BRADFORD  TITCHENER 

M.  A.  (Oxon.),  Ph.  D.  (Leipzig), 
LL.  D.  (University  of  Wisconsin), 

Member  of  the  Aristotelian  Society  and  of  the  Neurological   Society 

of  London ;  Associate  Editor  of  Mind  and  of  the  American 

Journal  of  Psychology  ;  Saae  Professor  of  Psychology 

in  the  Cornell  University. 


AN  OUTLINE  OF  PSYCHOLOGY 

Third  edition,  revised  and  enlarged.    Cloth,  8vo,  $1.50  net. 

A  PRIMER  OF  PSYCHOLOGY 

Third  edition,  revised  and  enlarged.    Cloth,  8vo,  $1.00  net. 

EXPERIilENTAL  PSYCHOLOGY,  QUALITATIVE : 

student's  Manual.    Second  edition,  revised.    Cloth,  8vo,  $1.00  net. 
Instructor's  Manual.    Cloth,  8vo,  $2.50  net. 

TRANSLATIONS 

O.  KUELPE,  Outlines  of  Psychology.  Translated  by  E.  B. 
TiTCHENER.     Large  8vo,  ^2.60  net. 

O.  KUELPE,  Introduction  to  Philosophy.  Translated  by 
W.  B.  PiLLSBURY  and  E.  B.  Titchener.     8vo,  ^1.60  net. 

"W.  WTJNDT,  Lectures  on  Human  and  Animal  Psy- 
chology. Translated  by  J.  E.  Creighton  and  E.  B. 
Titchener.     Second  edition,  revised.     Large  8vo,  ^2.60  net. 

W.  WTJNDT,  Ethics :  An  Investigation  of  the  Facts  and 
Laws  of  the  Moral  Life.  Translated  by  J.  H.  Gulliver, 
E.  B.  Titchener,  and  M.  F.  Washburn.  Vol.  I,  The  Facts 
of  the  Moral  Life.  Large  Svo,  ^2.25  net.  Vol.  II,  Ethical 
Systems.  $1.75  net.  Vol.  Ill,  The  Principles  of  Morality 
and  the  Sphere  of  their  Validity.     I2.00  net. 

W.  "WUNDT,  Principles  of  Physiological  Psychology. 
Translated  by  E.  B.  Titchener.  Vol,  I,  Introduction  :  The 
Bodily  Substrate  of  the  Mental  Life.     Large  Svo,  I3.00  net. 


THE  MACMILLAN  COMPANY 

66  FIFTH  AVENUE,  NEW  YORK 


*- 


EXPERIMENTAL  PSYCHOLOGY 

A  Manual  of  Laboratory  Practice 

BY 

EDWARD  BRADFORD  TITCHENER 


VOLUME  II 

QUANTITATIVE    EXPERIMENTS 
PART  I.    STUDENT'S  MANUAL 


Eine  Untersuchung,  die  an  sich  nicJit  schwierig  ist, 
aber  Geduld^  Aufmerksamkeit,  Ausdauer  und  Treue 
erfodert. — Fechner. 


THE    MACMILLAN  COMPANY 

LONDON:  MACMILLAN   &   CO.,   Ltd. 
1905 

All  rights  reserved  \  v) 


Copyright,  1905, 
By  the  MACMILLAN  COMPANY 

Set  up  and  electrotyped.     Published  October,  1905. 


QP  3S-J 

T  S3 

/  90S 


PREFATORY     NOTE:     SUGGESTIONS 
TO    STUDENTS 

The  Psychological  Experiment,  Qualitative  and  Quantitative — The  Quantitative 
Experiment  in  Practice — Laboratory  Partnerships — Note  Book,  Essay  Book,  Com- 
monplace Book — The  Modern  Languages. 

The  Psychological  Experiment,  dualitative  and  duantitative. — 

The  object  of  the  qualitative  experiment  in  psychology  is— if  we 
may  sum  it  up  in  a  single  word — to  describe ;  the  object  of  the 
quantitative  experiment  is  to  measure.  In  the  former,  we  seek 
to  gain  familiarity,  by  methodically  controlled  introspection, 
with  some  type  or  kind  of  mental  process ;  we  Hve  through,  at- 
tentively and  in  isolation,  some  special  bit  of  mental  experience, 
and  then  give  in  words  a  report  of  the  experience,  making  our 
report,  so  far  as  possible,  photographically  accurate.  Numerical 
determinations,  formulae,  measurements,  come  into  account  only 
in  so  far  as  they  are  necessary  to  the  methodical  control  of  in- 
trospection ;  they  are  not  of  the  essence  of  the  experiment.  The 
question  asked  of  consciousness  is  the  question  '  What  ? '  or 
*  How  ? '  :  What  precisely  do  I  find  here,  in  this  attentive  con- 
sciousness }  How  precisely  is  this  fusion  put  together  ? — In  the 
quantitative  experiment,  on  the  other  hand,  we  make  no  attempt 
at  complete  description  :  it  is  taken  for  granted  that  the  mental 
processes  now  under  examination  have  become  familiar  by  prac- 
tice. What  we  do  is  to  carry  out  a  long  series  of  observations, 
under  the  simplest  and  most  general  introspective  conditions. 
Then  we  gather  up  the  results  of  these  observations  in  mathemati- 
cal shorthand,  and  express  them  numerically  by  a  single  value. 
The  questions  asked  of  consciousness  are,  in  the  last  analysis,  two 
only :  *  Present  or  absent  t '  and  *  Same  or  different  t '  For  in- 
stance :  Do  you  still  hear  a  tone }  or :  Is  this  weight  heavier 
than  this  other,  or  lighter  than  it,  or  just  as  heavy?  When,  as 
we  have  said,  a  great  many  observations  of  this  sort  have  been 


4195^ 


vi  Prefatory  Note  :  Suggestions  to  Students 

taken,  the  whole  set  of  results  is  thrown  into  quantitative  form. 
On  the  average,  we  can  still  hear  a  tone  of  so-and-so  many  vibra- 
tions ;  on  the  average,  we  can  distinguish  two  weights  if  they  dif- 
fer by  such-and-such  an  amount.  The  question  which  the  quan- 
titative experiment  answers  is,  therefore,  some  variant  of  the 
question  *  How  much  ? '  Notice,  however,  that  this  is  not  the 
question  asked  of  consciousness.  That  question  is  always  the  one 
or  other  of  the  two  just  mentioned:  Present  or  absent.?  and 
Same  or  different }  Here,  then,  is  a  second  difference  between 
the  qualitative  and  the  quantitative  experiment.  The  former, 
aiming  at  description,  comes  to  an  end  when  introspection  has 
made  its  report ;  the  latter,  aiming  at  measurement,  subjects 
the  results  of  introspection  to  mathematical  treatment.  The  ex- 
periments are  complementary,  each  sacrificing  something  and 
each  gaining  something.  The  qualitative  experiment  shows  us 
all  the  detail  and  variety  of  the  mental  life,  and  in  so  doing  for- 
bids us  to  pack  its  results  into  formulae ;  the  quantitative  experi- 
ment furnishes  us  with  certain  uniformities  of  the  mental  life, 
neatly  and  summarily  expressed,  but  for  that  very  reason  must 
pass  unnoticed  many  things  that  a  qualitatively  directed  intro- 
spection would  bring  to  light. 

The  Quantitative  Experiment  in  Practice. —  In  general,  the 
rules  for  the  conduct  of  a  quantitative  experiment  are  the  same 
as  those  for  a  qualitative  experiment  (vol.  I.,  xiii.  f.).  There  are, 
however,  in  practice,  certain  well-marked  differences  between  the 
two  types  of  experiment. 

(i)  In  the  first  place,  the  quantitative  experiment  demands 
much  more  *  outside  '  preparatory  work  than  does  the  qualitative. 
Most,  if  not  all,  of  the  reading  done  in  preparation  for  the  ex- 
periments of  vol.  I.  could  be  done  within  the  laboratory.  This 
is  not  the  case  with  the  experiments  that  you  are  now  to  perform. 
The  quantitative  experiment  sums  up,  in  a  single  representative 
value,  the  results  of  a  large  number  of  observations.  It  is  clear, 
then,  that  the  conditions  of  observation  must  be  the  same 
throughout :  otherwise  the  results  will  not  be  comparable.  The 
observations  must  be  arranged,  distributed,  timed,  spaced,  varied, 
repeated,  upon  a  definite  plan  ;  and  the  plan  itself  must  be  laid 
out  with  a  view  to  the  object  and  materials  of  the  particular  ex- 


Prefatory  Note :    Suggestions  to  Students  vii 

periment.  This  means  that  you  must  know  your  method ;  you 
must  have  made  out  a  complete  scheme  of  work  before  you  en- 
ter the  laboratory. 

It  is  not  necessary  that  (?,  as  well  as  E^  know  the  method  to  be  employed 
in  a  given  experiment ;  if  E  has  made  out  his  plan,  he  can  give  such  in- 
struction to  O  as  shall  secure  the  object  at  which  the  method  aims.  It  is, 
indeed,  advisable  that  the  members  of  a  laboratory  partnership,  at  the  be- 
ginning of  the  term,  take  two  different  methods  for  study,  so  that  each 
serves  as  O  in  an  experiment  with  the  method  of  which  he  is  not  familiar. 
For  even  if  these  two  experiments  can  be  done  but  once,  still,  E  and  O  gain 
a  wider  experience  than  they  would  obtain  from  the  repetition  of  one  and  the 
same  experiment ;  while  if  they  can  be  repeated, — as  they  should  be,  with 
reversal  of  function,  whenever  time  allows, — each  student  has  as  E  the 
advantage  of  the  experience  which  he  gained  as  O. 

(2)  Again  :  the  quantitative  experiment  demands  a  more  sus- 
tained attention  than  the  qualitative.  Many  of  the  experiments  of 
vol.  I.  could  be  dropped,  as  soon  as  O  showed  signs  of  fatigue,  and 
resumed,  in  the  next  laboratory  session,  practically  at  the  point 
at  which  they  had  been  left.  Some  of  them  actually  called  for  a 
set  of  separate  observations,  so  that  the  work  naturally  fell  into 
short  periods,  with  rest-pauses  between.  But  we  cannot  drop  a 
method,  half  finished,  and  take  it  up  again  at  a  later  time.  Once 
begun,  the  quantitative  experiment  must  be  carried  through  to 
its  proper  end.  We  make  out  the  plan  of  the  experiment  be- 
forehand, in  order  that  our  results  may  be  comparable  and  homo- 
geneous ;  we  must,  then,  and  for  the  same  reason,  adhere  to  the 
■plan  in  practical  work. 

(3)  Lastly,  the  service  of  the  apparatus  is,  in  general,  a  more 
delicate  matter  in  the  quantitative  than  it  is  in  the  qualitative  ex- 
periment. In  some  cases,  this  is  due  to  the  greater  refinement 
and  complication  of  the  instruments  themselves.  For  the  most 
part,  however,  we  employ  simple  instruments  ;  and  the  demands 
of  care  and  accuracy  laid  upon  E  derive  from  that  main  source  of 
difference  between  these  and  our  earlier  experiments,  the  obliga- 
tions of  method.  The  disposition  of  apparatus  in  the  second, 
third,  fourth  series  must  be  exactly  as  it  was  in  the  first  series ; 
else  the  results  of  the  different  series  cannot  be  grouped  together 
to  yield  a  single  result.     But  further  :  method-work  in  psychology 


viii  Prefatory  Note :  Suggestions  to  Students 

makes  certain  claims  upon  the  apparatus  which  qualitative  work 
does  not ;  claims  which  we  cannot  here  specify,  but  which  will 
be  pointed  out  as  the  experiments  proceed. — 

In  fine,  then,  there  is  preparatory  work  to  be  done  outside  the 
laboratory  ;  and  in  the  laboratory  O  must  be  steadily  and  uniformly 
attentive,  while  E  has  to  carry  the  plan  of  the  method  and  at  the 
same  time  to  watch  his  apparatus.  Does  not  this  mean  that 
quantitative  experiments  in  psychology  are  more  difficult  than 
qualitative  }  On  the  whole,  yes.  There  are,  however,  two  points 
that  you  should  keep  in  mind.  The  first  is  that  there  are  degrees 
of  difficulty  in  quantitative,  as  there  are  in  qualitative  work. 
Several  of  the  experiments  prescribed  in  this  volume  are  easier 
than  experiments  prescribed  in  vol.  I.  And  the  second  is  that 
you  will  bring  to  bear  upon  the  difficulties  of  this  part  of  the 
Course  all  the  experience  that  you  have  acquired  in  the  first  part. 
The  difficulty  of  passing  from  qualitative  to  quantitative  work  is, 
in  all  probability,  nothing  like  as  great  as  the  difficulty  which 
you  faced  on  first  beginning  the  Course, — when  you  were  called 
upon  to  renounce  popular  psychology,  to  take  up  an  entirely  new 
attitude  to  mind,  to  examine  consciousness  at  first-hand  by  intro- 
spection. So  that,  while  the  work  becomes  increasingly  difficult, 
there  is  nothing  to  be  afraid  of. 

Laboratory  Partnerships. — A  few  of  the  following  experiments 
can  be  performed  by  a  single  student ;  but  the  great  majority  re- 
quire the  cooperation  of  two  students,  E  and  O.  It  is  very  de- 
sirable that  the  partnership  formed  for  the  qualitative  work,  if  it 
was  congenial,  be  continued  during  this  second  part  of  the  Course. 
If,  however,  your  former  partner  is  not  completing  the  Course,  or 
if  you  are  yourself  resuming  the  work  after  an  interval,  it  is 
desirable  that  you  choose  your  own  partner,  rather  than  leave  the 
choice  to  the  Instructor.  E  and  O  must  be  in  sympathy,  and 
must  have  full  confidence  in  each  other,  if  the  quantitative  ex- 
periments are  to  be  successfully  carried  out. 

It  may  be,  of  course,  that  you  cannot  make  such  an  arrange- 
ment as  is  here  suggested,  and  that  your  partner  is  selected  by 
the  Instructor.  If,  now,  he  does  not  prove  to  be  a  congenial  as- 
sociate, see  to  it  that  the  partnership  is  promptly  dissolved,  and 
an  exchange  made.     Do  not  hold  back  for  fear  of  offence,  or  for 


Prefatory  Note  :  Suggestioits  to  Students  ix 

any  false  shame  about  making  a  complaint.  You  may  be  pretty 
sure  that,  if  you  do  not  get  on  with  him,  neither  does  he  get  on 
with  you.  Talk  the  situation  over  with  him,  and  part  by  mutual 
consent.     It  is  foolish  to  spoil  your  work  for  a  scruple. 

On  your  own  side,  try  hard  to  live  up  to  the  terms  of  the 
laboratory  partnership  as  described  in  vol.  I.,  xvi. 

Note  Book,  Essay  Book,  Commonplace  Book. — The  note  book  is 
to  be  kept  just  as  for  the  qualitative  experiments  (vol.  I.,  xvi.  f.), 
with  such  minor  differences  of  record  as  the  changed  character 
of  the  work  makes  necessary.  It  should  be  handed  in  to  the 
Instructor  immediately  after  the  writing  up  of  each  separate  ex- 
periment, so  that  criticisms  and  corrections  may  be  made  while 
the  performance  of  the  experiment  is  still  fresh  in  your  memory. 

From  time  to  time,  as  the  Course  proceeds,  you  will  be  asked 
to  write  an  essay  upon  some  general  topic  connected  with  the 
experiments.  Write  the  essays  in  a  second  note  book,  kept 
specially  for  them.  Write  upon  one  side  of  the  paper  only,  but 
enter  your  references  and  footnotes  upon  the  other,  blank  side. 
If  you  have  occasion  to  quote  a  foreign  author,  give  a  transla- 
tion of  the  quoted  sentences  in  the  body  of  the  essay,  and  write 
out  the  original  upon  the  corresponding  blank  page  of  the  essay 
book.  Never  shirk  the  labour  of  translation  ;  and,  for  accuracy's 
sake,  never  omit  the  transcription  of  the  original  passage. 

If  you  intend  to  take  a  further  course  in  Systematic  Psychology 
at  the  conclusion  of  this  laboratory  work  :  still  more,  if  you  in- 
tend to  pursue  graduate  studies  in  psychology  :  it  will  be  well 
worth  your  while  to  keep,  besides  the  note  book  and  the  essay 
book,  a  third  book  for  the  recording  of  miscellaneous  psychologi- 
cal notes.  This  commonplace  book  might  contain,  e.  g.,  abstracts 
of  the  authors  read  for  an  essay,  lists  of  problems  that  suggest 
themselves  to  you  in  the  course  of  the  laboratory  exercises,  criti- 
cisms of  current  theories,  theoretical  ideas  of  your  own,  lists  of 
references  upon  topics  in  which  you  are  interested,  odd  bits  of 
psychologising  that  you  meet  with  in  your  other  laboratory  and 
class-room  work  or  in  your  general  reading,  apt  illustrations  of 
psychological  laws  that  you  come  across  in  everyday  life,  the 
questions  and  objections  hurled  at  you  by  outside  critics,  etc.,  etc. 
To  the  psychologist,  the  whole  of  *  this  great  and  glorious  world ' 


X  Prefatory  Note:  Suggestions  to  Students 

is  matter  for  psychology :  and  if  you  keep  your  commonplace 
book  diligently  for  a  few  months,  you  will  be  surprised  to  find 
how  much  psychology  there  is  in  life,  and  how  close  and  mani- 
fold are  the  connections  between  the  laboratory  and  your  experi- 
ences beyond  the  laboratory.  You  will  also  be  surprised  to  find 
how  naturally  and  in  how  real  a  sense  the  keeping  of  the  com- 
monplace book  leads  you  to  psychologise  for  yourself. 

The  Modern  Languages. — It  has  been  assumed,  throughout  this 
Course,  that  the  student  is  able  to  read  psychological  works  in 
French  and  German.  It  is,  indeed,  necessary  to  make  this  as- 
sumption, if  the  Course  is  to  be  regarded  at  all  seriously.  For 
experimental  psychology  was  in  origin  a  German  science,  and 
has  become  international.  The  classical  treatises  of  the  earlier 
period  are  in  German,  and  have  not  been  translated.  Nor  have, 
e.  g.f  those  critical  essays  by  J.  Delboeuf  which  suggest  the  theory 
of  mental  measurement  adopted  in  this  book. 

It  is  probable,  then,  that  the  first  essay  which  you  are  called 
upon  to  write  in  connection  with  the  Course  will  demand  a  read- 
ing of  French  and  German,  as  well  as  of  English,  sources.  If 
you  are  familiar  with  the  foreign  languages,  well  and  good.  If 
you  are  not,  do  not  on  that  account  lose  heart,  or  ask  for  another 
assignment.  Set  to  work,  on  the  basis  of  your  school  knowledge, 
and  read  the  references.  You  are  not  required,  by  this  Course, 
to  become  a  fluent  linguist,  or  to  acquire  a  literary  appreciation 
of  French  and  German  style :  you  are  required  simply  to  read 
and  understand  certain  prose  passages  which  have  a  technical 
and  therefore  a  limited  vocabulary.  If  needs  must,  take  a  few 
private  lessons  in  the  less  familiar  language  from  some  graduate 
student  who  knows  both  the  language  and  psychology.  At  any 
rate,  make  up  your  mind  that  by  the  end  of  the  term  you  are 
going  to  read  French  and  German  psychology  with  accuracy  and 
with  some  ease.  The  task  is  much  less  formidable  than  it  looks, 
and  every  half-hour  spent  upon  its  counts. 


TABLE  OF  CONTENTS 

INTRODUCTION 
MENTAL  MEASUREMENT 

PAGE 

§   I.  Measurement xix 

§  2.  Mental  Measurement       .......  xxi 

§  3.  An  Analogy xxvii 

§  4.  Three  Problems  of  Quantitative  Psychology     .         .         .  xxix 

§  5.  Some  Technical  Terms xxxvii 

§  6.  The  Further  Programme  of  Quantitative  Psychology         .  xxxviii 

§  7.  Questions .  xl 

CHAPTER  I.     PRELIMINARY  EXPERIMENTS 

Experiment  I 
§  8.   The  Qualitative  RL  for  Tones :  the  Lowest  Audible  Tone       .        i 

Experiment  II 
§  9.   The  Qualitative  ^Z  for  Tones  :  Alternative  Experiment  .       10 

Experiment    III 
}   10.    The  Qusditative  TV?  for  Tones  :  the  Highest  Audible  Tone      .       12 

Experiments  IV-VI 
\  II.   The  Intensive  7?Z  for  Pressure 14 

Experiments  VII,  VIII 

\  12.   The  Intensive  RL  for  Sound .22 

j   13.   Weber's  Law 25 

Experiments  IX-XII 
)   14.    Demonstrations  of  Weber's   Law:    Fechner's  Cloud  Experi- 
ment;   Fechner's  Shadow  Experiment;    Sanford's  Weight 
Experiment;  Ebbinghaus'  Brightness  Experiment        .         .       29 
xi 


xii  Table  of  Contents 

CHAPTER  II.     THE  METRIC  METHODS 
§   15.    The  Law  of  Error     .         . 38 

Experiments  XIII,  XIV 
§   16.   The  Method  of  Limits  (Method  of  Just  Noticeable  Differences : 
Method     of     Least     Differences :      Method     of     Minimal 
Changes):    Determination   of   the  DL  for  Brightness  and 
Tone    . 55 

Experiment  XV 
§   17.    Fechner's  Method  of  Average  Error  (Method  of  Reproduction : 
Method  of  Adjustment  of  Equivalent  R) :  the  Equation  of 
Visual  Extents 70 

Experiments  XVI,  XVII 
§   18.   The  Method  of  Equivalents:    Application  of  the  Method  to 

Cutaneous  Extents  and  to  Pressure       .         .         ,         .         .       TJ 

Experiments   XVIII,  XIX 
§   19.    The    Method  of    Equal    Sense- Distances   (Method  of    Mean 
Gradations :    Method  of  Supraliminal  Differences :  Method 
of  Equal-appearing  Intervals) :  Application  of  the  Method  to 
Intensities  of  Sound  and  to  Brightness  .         .         .         .81 

Experiment  XX 
§  20.   The  Method  of  Constant  R  (Method   of   Right  and  Wrong 
Cases):    Determination  of   the  Limen  of   Dual  Impression 
upon  the  Skin 92 

Experiment  XXI 
§  21.    The   Determination  of  Equivalent  R  by  the  Method  of  Con- 
stant R :  the  Comparison  of  Movements  of  the  Arm     .         .104 

Experiments  XXII,  XXIII 
§  22.   The  Method  of  Constant  ^-Differences  (Method  of  Right  and 
Wrong  Cases) :    Determination  of  the  DL  for  Intensity  of 
Sound  and  for  Lifted  Weights 106 


Table  of  Contents  xiii 

CHAPTER  III.     THE  REACTION  EXPERIMENT 

§  23.   The  Electric   Current  and  the  Practical  Units  of  Electrical 

Measurement 120 

§  24.    The  Technique  of  the  Simple  Reaction 141 

Experiment  XXIV 
§  25.   The  Three  Types  of  Simple  Reaction       .  "      .         .         .         .167 

Experiment  XXV 
§  26.   Compound  Reactions:  Discrimination,  Cognition  and  Choice  .     185 

Experiment  XXVI 
§  27.    Compound  Reactions :  Association 192 

CHAPTER  IV.     THE  PSYCHOLOGY  OF  TIME 

Experiment  XXVII 
§  28.    The  Reproduction  of  a  Time  Interval 196 

List  of  Materials 199 

Index  of  Names  and  Subjects     .        .        .        .        •        .        .    201 


INDEX  OF  FIGURES 

FIG.  PAGE 

1.  Illustration  of  Mental  Measurement xxxiii 

2.  Appunn's  Lamella I 

3.  Gal  ton's  Whistle 12 

4.  Edelmann's  Galton  Whistle 13 

5.  Scripture's  Touch  Weights 15 

6.  von  Frey's  Hair  ^Esthesiometer 16 

7.  Application  of  Hair  to  the  Cutaneous  Surface  (von  Frey)     .         .  16 

8.  von  Frey's  Limen  Gauge 17 

9.  Regulating  Apparatus  for  Limen  Gauge       .         .         .         .  18,  19 

10.  Lehmann's  Acoumeter          ........  24 

11.  Correlation  of  Intensive  Pressure  Sensation  with  its  Adequate 

Stimulus  (Hofler) 26 

12.  Correlation  of  Tonal  Pitch  with  Vibration  Rate  (Hofler)      .         .  27 

13.  Correlation  of  S  with  7?  under  Weber's  Law  (Wundt)           .         .  29 

14.  Arrangement  of  Apparatus  for  Fechner's  Shadow  Experiment     .  32 

15.  Discs  for  Kirschmann's  Photometer 36 

16.  Cattell's  Curve 42 

1 7.  Curves  of  Gauss'  Law  of  Error 43 

18.  Mechanical  Illustration  of  Cause  of  Curve  of  Frequency  (Galton)  46 

19.  Illustration  of  a  Normal  Distribution  (Galton)      ....  48 

20.  Tuning-forks  with  Rider  Attachments 6S 

21.  Tuning-forks  with  Screw  Attachments 68 

22.  Galton  Bar 76 

23.  Sound  Pendulum          .         .         . 81 

24.  Delbceuf's  Disc  :  First  Pattern 88 

25.  Delbceuf's  Dark  Box 89 

26.  Delbceuf's  Disc :  Second  Pattern 89 

27.  Arrangements  of  Discs    for   the  Method  of  Mean  Gradations 

(Wundt) 90 

28.  Wundt's  Triple  Colour  Mixer       .......  90 

29.  Miiller's  Curve  of  the  Distribution  of  the  J^L      ....  94 

XV 


xvi  Index  of  Figures 

30.  Miinsterberg's  Apparatus  for  the  Comparison  of  Movements  of 

the  Arm  ...........  105 

31.  Miiller's  Curve  of  the  Distribution  of  the  Z>Z       .         .         .         .107 

32.  Carrier  Bracket  for  Lifted  Weights 116 

33.  Simple  Voltaic  Cell 122 

34.  Arrangements  of  Primary  Batteries 124 

35.  Four-frame  Nichols  Rheostat 12S 

36.  Schema  of  Lamp  Battery 131 

37.  Wright-Scripture  Lamp  Battery 132 

38.  Ammeter  (Bottone) 133 

39.  Simple  Forms  of  Ammeter  and  Voltmeter 13^ 

40.  Schema  of  Measurement  of  Lamp  Resistance       .         .         .         -133 

41.  Wheatstone  Bridge 135 

42.  Metre  Bridge  (S.  P.  Thompson) 136 

43.  Schemata  of  Direct  Current  and  Alternating  Current  Dynamos 

(Glazebrook) 13; 

44.  Schemata  of  Series-wound  and  Shunt-wound  Dynamos         .         .13^ 

45.  The  Hipp  Chronoscope  (front) 142 

46.  The  Hipp  Chronoscope  (back) 142 

47.  Arrangement  I  of  the  Simple  Reaction  Experiment      .         .         •  I4f 

48.  Arrangement  II  of  the  Simple  Reaction  Experiment   .         .         .  14^ 

49.  Arrangement  III  of  the  Simple  Reaction  Experiment           .         .  147 

50.  PohPs  Commutator 14^ 

51.  Direction  of  Currents  through  Pohl's  Commutator        .         .         •  HS 

52.  Simple  Control  Hammer  (Wundt)         .         .         .         .         .         .15^ 

53.  Noise  Stimulator 153 

54.  Sound  Hammer  (Wundt) 15^ 

55.  Martins'  Arrangement  for  Simple  Reaction  to  Clangs  .         .         •  iSi 

56.  Schema  of  the  Wundt-Lange  Optical  Pendulum  .         .         .         .15^ 

57.  Krille's  Interruptor 15; 

58.  Cattell's  Arrangement  for  Voice  Reactions  to  Visual  Stimuli           .  15J 

59.  Jastrow's  Shutter I5( 

60.  Scripture's  Touch  Key I5< 

61.  Dessoir's  Sensibilometer i6c 

62.  Arrangement    for    Simple    Reaction     to     Electric    Stimulation 

(Dumreicher) i6c 

63.  von  Vintschgau's  Thermophor 16] 

64.  Kiesow's  Temperature  Stimulator 161 


Index  of  Figures 


xvu 


Reactions 


65.  Moldenhauer's  Smell  Stimulator 

66.  Buccola's  Smell  Stimulator 
6-].  Zvvaardemaker's  Arrangement  for  Olfactory 

68.  Reaction  Key  (Telegraph  Model) 

69.  Jastrow's  Reaction  Key 

70.  Scripture-Dessoir  Reaction  Key  . 

71.  Cattell's  Lip  Key  .... 

72.  Jastrow's  Speech  Key 

73.  Galvanometer  (Peyer  et  Favarger) 

74.  Two-way  Switch  .... 

75.  Rheochord  for  Hipp  Chronoscope 

76.  Distribution  of  Apparatus  in  Arrangement  II 

77.  Distribution  of  Apparatus  in  Arrangement  I 

78.  Frequency  Polygon  of  '  Reaction  Times '  of  Frog  (Yerkes) 

79.  Frequency  Polygon  of  Reaction  Times  (Thorndike's  data) 

80.  Frequency  Polygon  of  Reaction  Times  (Thorndike's  data) 

8 1 .  Frequency  Polygon  of  Reaction  Times  (Alechsieff) 

82.  Reading  Telescope  (Cambr.  Inst.  Co.) 

83.  Five-finger  Reaction  Keys  (Zimmermann)     . 

84.  Roemer's  Exposure  Apparatus     .... 

85.  Vierordt's  Lever 


162 
163 
163 
164 
164 
16s 
165 
166 
168 
168 
168 
169 
174 
179 
180 
180 
181 
188 
189 

193 
196 


INTRODUCTION  :  MENTAL  MEASUREMENT 

§  I.  Measurement. — Whenever  we  measure,  in  any  depart- 
ment of  natural  science,  we  compare  a  given  magnitude  with 
some  conventional  unit  of  the  same  kind,  and  determine  how 
many  times  the  unit  is  contained  in  the  magnitude.  Let  P  be 
the  magnitude  to  be  measured,  and/  the  unit  in  terms  of  which 
it  is  to  be  expressed.  The  result  of  our  measurement  of  P  is 
the  discovery  of  the  numerical  ratio  existing  between  P  and  /. 
We  state  this  result  always  in  terms  of  an  equation  :  P=z-p. 
The  object  of  measurement,  then,  is  the  giving  of  such  values  to 
X  and  y  that  the  equation  may  be  true. 

When  we  say,  e.  g.,  that  Mt.  Vesuvius  is  4200  ft.  high,  we 
mean  that  the  given  linear  magnitude  P,  the  distance  from  sea 
level  to  the  topmost  point  of  the  volcano,  contains  the  conven- 
tional unit  of  linear  measurement,  i  ft.,  four  thousand  two  hun- 
dred times  :  P  ^^-^  p.  When  we  say  that  an  operation  lasted 
40  minutes,  we  mean  that  the  given  temporal  magnitude,  the  time 
occupied  by  the  operation,  contained  the  conventional  time  unit, 
I  min.,  forty  times  :  P  =1^-^-  p.  When  we  say  that  an  express 
package  weighs  three  quarters  of  a  pound,  we  mean  that  the 
package,  laid  in  the  scale  pan,  just  balances  the  sliding  weight 
when  this  is  placed  at  the  twelfth  short  stroke  beyond  the  first 
long  stroke  or  zero  point  of  the  bar, — the  distance  between  any 
two  long  strokes  giving  the  conventional  unit  i  lb.,  and  the  dis- 
tance between  any  two  short  strokes  the  conventional  sub-unit 
I  oz.  =r  tV  lb.  :  so  that  P  =  |  /  in  terms  of  lb.,  ox  P=^^  p'm 
terms  of  oz.  In  the  same  way,  we  might  say  that  the  height  of 
Mt.   Vesuvius  is  ||f^  mile  ;  or  that  the  time  of  the  operation 

is  4  0.    V.|. 

These  instances  show — what  must  always  be  borne  in  mind — 


XX  Introduction:  Mental  Measurement 

that  the  unit  of  measurement  is  conventional.  Its  choice  is  sim- 
ply a  matter  of  practical  convenience.  Scientific  men  are  now 
generally  agreed  that  the  unit  of  space  shall  be  the  i  cm.,  the  unit 
of  time  the  i  sec,  and  the  unit  of  mass  the  i  gr.  ;  so  that  the  unit 
of  mechanical  energy  is  the  amount  contained  in  a  body  of  i  gr. 
moving  through  i  cm.  in  i  sec.  There  is,  however,  nothing  ab- 
solute and  nothing  sacrosanct  about  these  units.  Our  measure- 
ments would  be  every  bit  as  valid,  just  as  much  true  measure- 
ments, if  we  took  as  units  the  pace  or  span  or  cubit,  the  average 
time  of  a  step  in  walking  or  of  a  respiratory  movement,  the  ounce 
or  pound.  The  metric  system  makes  calculation  easy,  relates 
the  three  fundamental  quantities  in  a  very  simple  manner :  but 
that  is  its  sole,  as  it  is  its  sufficient  claim  to  acceptance. 

The  prototype  of  all  measurement  is  linear  measurement  in 
space.  We  can  literally  superpose  one  portion  of  space  upon 
another,  for  purposes  of  measurement :  we  can  hold  the  compared 
portions  together  for  as  long  a  time  as  we  like ;  we  can  shift  the 
one  portion  to  and  fro  upon  the  other.  The  linear  space  unit 
is  thus  the  most  easily  manipulated  of  all  units  of  measurement. 
Hence  there  is  a  tendency,  in  natural  science,  to  reduce  all 
quantitative  comparisons  to  the  comparison  of  spatial  magnitudes. 
We  compare  masses,  with  the  metric  balance,  by  noting  the  de- 
flections of  the  pointer.  We  measure  the  intensity  of  an  electric 
current  by  noting  the  deflections  of  the  galvanometer  needle. 
We  measure  rise  and  fall  of  temperature  by  the  rise  and  fall  of 
mercury  in  the  thermometer.  We  determine  the  period  of  a  tun- 
ing fork  by  the  graphic  method.  It  is,  moreover,  with  spatial 
measurements  that  we  are  chiefly  concerned  in  everyday  life. 
We  are  all,  to  some  extent,  practised  in  the  estimation  of  space 
magnitudes,  however  helpless  we  may  be  when  called  upon  to 
grade  weights  or  to  estimate  brightnesses. 

Finally,  it  is  to  be  noticed  that  every  measurement  implies 
three  terms,  expressed  or  understood.  In  measuring  the  height 
of  Mt.  Vesuvius  we  had  the  zero  point,  or  sea  level  ;  the  highest 
point  of  the  mountain ;  and  the  point  lying  i  ft.  above  sea  level 
or  I  ft.  below  the  highest  point.  In  measuring  the  time  of  the 
operation,  we  had  the  beginning  and  end  of  the  period  occupied 
by  it,  and  the  time  point  lying  i  min.  (or  i  hr.)  distant  from 


§   2.     Me7ital  Measurement  xxi 

the  beginning.  In  weighing  our  package,  we  had  the  zero  point 
upon  the  scale  bar  ;  the  Hmiting  point  at  which  the  sliding  weight 
was  just  counterbalanced  ;  and  the  mark  that  lay  i  oz.  (or  i  lb.) 
from  the  zero  point.  There  are  various  devices — the  introduc- 
tion of  submultiples  of  the  unit,  the  use  of  the  vernier — for  in- 
creasing the  accuracy  of  measurement ;  there  are  other  devices 
for  standardising  the  conditions  (temperature,  stress)  under 
which  a  measurement  is  made  ;  there  are  mathematical  rules  for 
calculating  the  *  probable  error '  of  a  given  measurement.  These 
are  all  refinements  of  the  art  of  measuring.  The  essential  thing 
is  that  we  have  our  three  terms  :  the  limiting  points  of  the  mag- 
nitude to  be  measured,  and  a  point  lying  at  unit  distance  from: 
the  one  or  the  other  limiting  point. 

The  third  term,  without  which  measurement  is  impossible, 
need  not,  however,  be  expressed.  Suppose  that  two  black  strokes 
are  made  upon  a  sheet  of  paper,  and  that  you  are  asked  to  say 
how  far  the  one  is  above  the  other,  or  to  the  right  of  the  other. 
You  reply,  without  difficulty,  "Two  inches"  or  "Five  centi- 
metres." But  that  means  that  you  have  mentally  introduced  a 
third  term  :  the  unit  mark,  inch  or  cm.,  with  which  you  have  be- 
come familiar  in  previous  measurements.  Without  this,  you 
could  only  have  said :  "  The  one  mark  is  above  the  other  "  or 
"to  the  right  of  the  other"  ;  you  could  not  have  answered  the 
question  "how  far."  How  long  is  a  given  stretch  of  level  road  } 
Two  hundred  and  fifty  yards .?  A  quarter  of  a  mile }  Most 
people  have  no  mental  unit  for  such  a  measurement.  Either 
they  say  "  It  looks  about  as  long  as  from  so-and-so  to  so-and-so," 
— comparing  it  with  a  familiar  distance  ;  or  they  make  a  rough 
determination  by  pacing  the  distance  itself.  How  deep  is  this 
well }  Very  few  people  can  say,  even  if  they  can  see  the  water. 
So  a  stone  is  dropped  in,  and  the  seconds  are  counted  until  the 
splash  is  heard.  The  pace  or  the  familiar  distance  gives  a  third 
term  for  the  measurement  of  the  road ;  and  we  know  that  the 
distance  traversed  by  the  stone  in  falling  is  the  product  of  the 
distance  traversed  in  the  first  second  (about  490  cm. — our  third 
term)  into  the  square  of  the  time.  Where  there  is  no  such  third 
term,  there  is  no  measurement.      This  rule  is  universal. 

§  2.  Mental  Measurement. — There  can  be  no  question  but  that, 


xxii  Introduction  :   Mental  Meas?irement 

in  some  way  or  other,  mental  processes  are  measurable.  It 
would  be  strange,  indeed,  if  the  processes  of  the  physical  uni- 
verse, which  we  know  only  by  means  of  our  sense  organs  or  of 
instruments  which  refine  upon  our  sense  organs,  should  be  capa- 
ble of  measurement,  while  the  sensations  of  mental  science  were 
not :  if  all  measurement  in  the  physical  sciences  should  tend 
towards  spatial,  i.  e.^  visual  measurement,  while  yet  the  visual 
sensations  themselves  were  unmeasurable.  That  apart,  however, 
we  have  only  to  appeal  to  introspection  to  see  that  our  mental 
processes  furnish  the  raw  material  of  measurement.  Sensations 
differ  more  or  less  in  quality  :  a  given  tone  lies  higher  in  the  scale 
than  another,  a  given  green  is  more  yellow  than  another.  They 
differ  in  intensity  :  a  noise  may  be  louder  than  another  noise,  a 
brightness  stronger  than  another  brightness.  They  differ  in 
duration  :  the  taste  of  bitter  lasts  longer  than  the  taste  of  sweet, 
the  visual  after-image  lasts  longer  than  the  auditory.  They  differ 
in  extent :  one  red  is  larger,  one  pressure  spread  more  widely 
than  another.  These  differences  are  given  with  the  sensations  ; 
they  obtain  whether  or  not  we  know  anything  of  the  stimuli 
which  arouse  the  sensations ;  we  have  evidence,  in  the  history  of 
science,  that  they  were  remarked  and  utilised  long  before  the 
stimuli  were  known  or  measured.  But  if  we  have  differences  of 
more  and  less,  it  is  only  necessary  to  establish  the  unit,  the  third 
term,  in  order  to  convert  difference  into  measured  difference. 

This  establishment  of  the  unit  is,  however,  no  small  matter. 
We  have  said  that  the  units  of  physical  measurement  are  con- 
ventional. On  the  other  hand,  the  units  of  modern  physics  are 
accurate,  objective,  universal.  It  is  a  far  cry  to  these  units — the 
cm.,  the  sec,  the  gr. — from  such  things  as  the  day's  journey,  the 
barley-corn,  the  chaldron.  The  difficulty  of  choosing  the  unit, 
and  of  standardising  it  when  chosen,  is  much  greater  than  might 
at  first  thought  be  supposed,  and  can  be  fully  appreciated  only 
by  one  who  has  followed  historically,  step  by  step,  the  develop- 
ment of  scientific  theory  and  practice.  What  holds  in  this  regard 
of  physics  holds  also  of  psychology.  Moreover,  the  psychologist 
is  at  a  peculiar  disadvantage,  in  that  there  is  no  natural  unit  of 
mental  measurement.  The  human  body  affords  the  natural  units 
of  linear  measurement  :  foot,  pace,  cubit,  span.     The  height  of 


§   2.     Mental  Measurement  xxiii 

the  sun  in  the  heavens,  the  alternation  of  day  and  night,  the 
changes  of  the  moon, — these  are  all  natural  units  of  time  meas- 
urement. Units  of  weight  are  furnished  by  convenient  natural 
objects  (grain,  stone)  or  by  the  average  carrying  power  of  man 
or  animal  (pack,  load).  There  are  no  such  obvious  points  of 
reference  in  psychology.  Once  more  :  physics  is  able  to  relate 
and  combine  its  units,  to  reduce  one  to  another,  to  express  one 
in  terms  of  another ;  so  that  the  formula  for  mechanical  work, 
e.  g.,  has  the  form  ^^,  where  M  represents  mass  measurea 
in  gr.,  L  length  of  path  measured  in  cm.,  and  T  time  measured 
in  sec.  This  sort  of  interrelation  is  forbidden  by  the  very  nature 
of  mental  processes,  every  group  of  which  is  qualitatively  dissim- 
ilar to  every  other  group.  Hence  there  can  be  no  single  unit  of 
mental  measurement,  no  generalisation  of  the  units  employed 
in  special  investigations. 

Here,  then,  are  difficulties  in  plenty.  And  there  can  be  no 
question  but  that  these,  the  intrinsic  difficulties,  are  largely  re- 
sponsible for  the  tardy  advent  of  measurement  in  psychology,  and 
for  the  doubts  and  controversies  and  confusions  that  have  arisen 
since  the  methods  of  psychological  measurement  were  formulated. 
The  formulation  itself  dates  only  from  i860,  when  Gustav 
Theodor  Fechner  (i 801-1887),  gathering  together  scattered  ob- 
servations from  physics  and  astronomy  and  biology,  summing  up 
elaborate  investigations  of  his  own,  putting  his  physical,  mathe- 
matical and  psychological  knowledge  at  the  service  of  mental 
measurement,  published  his  Elemente  der  Psychophysik,  Fech- 
ner is  the  founder,  we  might  almost  say  the  creator,  of  quantita- 
tive psychology,  and  the  modern  student  who  will  understand  the 
principles  and  methods  of  mental  measurement  must  still  go  to 
school  with  Fechner. 

There  are,  however,  other  and  extrinsic  reasons  for  the  late 
development  of  a  quantitative  psychology.  If  we  are  to  have  a 
satisfactory  system  of  mental  measurements,  we  must  (as  will 
appear  later)  rely  largely  upon  the  results  of  physical  and  physio- 
logical research ;  and,  while  physics  has  been  securely  estab- 
lished for  some  centuries,  modern  physiology  may  be  said  to  date 
from  the  second  quarter  of  the  nineteenth  century.  Moreover, 
there  is  a  sharp  line  of  division  in  popular  thought — a  line  drawn 


xxiv  Introduction:  Mental  Measurement 

for  the  modern  world  by  Descartes — between  the  natural  and  the 
mental  sciences  ;  the  former  seem  to  be  quantitative  and  meas- 
urable, the  latter  qualitative  and  unmeasurable.  This  '  common 
sense '  point  of  view  has  all  the  weight  and  inertia  of  a  settled 
tradition.  Nay  more  :  at  the  beginning  of  the  nineteenth  cen- 
tury it  had  received  strong  reinforcement  from  the  philosophical 
side.  Immanuel  Kant  (1724- 1804) — not  a  mediaeval  philoso- 
pher, but  the  author  of  the  Critique  of  Pure  Reason  and  one  of 
the  most  influential  thinkers  of  modern  times — declared  roundly 
in  I  ']%6,  and  found  no  reason  later  to  change  his  opinion,  that 
psychology  could  never  attain  the  rank  of  a  true  science.  Some- 
thing was  done  towards  breaking  up  this  dogma  by  the  psycho- 
logical work  of  Johann  Friedrich  Herbart  (i  776-1 841).  Her- 
bart  is,  however,  as  weak  in  fact  as  he  is  strong  in  theory.  So 
that,  simple,  as  the  idea  of  mental  measurement  appears  to  us, 
who  come  after  Fechner,  we  need  not  read  far  in  the  psycholog- 
ical literature  of  the  early  nineteenth  century  to  realise  how  diffi- 
cult it  was,  before  Fechner,  even  seriously  to  entertain  the  idea, 
to  dwell  upon  it  as  practicable,  as  anything  more  than  a  bit  oi 
daring  speculation.  The  path  of  scientific  progress  is  littered 
with  brilliant  suggestions  :  it  is  so  easy  to  suggest,  and  so  hard 
to  grasp  the  suggestion,  to  work  it  out,  to  invent  the  methods  for 
turning  it  into  fact  !  Fechner  not  only  had  the  idea,  the  inspira- 
tion of  mental  measurement,  but  he  spent  ten  laborious  years  on 
its  actual  accomplishment. 

There  is  one  mistake,  so  natural  that  we  might  almost  call  it 
inevitable,  which  has  sorely  delayed  the  advance  of  quantitative 
psychology.  It  is  a  mistake  with  regard  to  the  object  of  meas- 
urement, the  mental  magnitude.  We  have  seen  that  every 
measurement  requires  three  given  terms ;  so  that  the  physical 
quantity  or  magnitude  is  not,  so  to  say,  a  single  term,  but  rather 
a  distance  between  terms,  a  section  of  some  stimulus  scale.  We 
are  apt  to  say,  carelessly,  that  we  have  measured  *  the  highest 
point '  of  Mt.  Vesuvius,  when  we  have  in  reality  measured,  in 
terms  of  our  arbitrary  unit,  the  distance  between  its  lowest  and 
highest  points.  It  is  not  the  point  that  is  the  magnitude,  but  the 
distance  between  points.     So  with   sensations  :  we  are   apt  to 


§   2.     Me7ital  Measurement  xxv 

think  of  a  brightness  or  a  tone  of  given  intensity  as  a  sensation 
magnitude,  as  itself  measurable.  Now  the  stimulus  is  measur- 
able :  we  can  measure,  in  terms  of  some  unit,  the  amplitude  of 
vibration  of  the  ether  or  air  waves  :  we  have  our  three  terms  to 
measure  with.  But  the  sensation,  the  brightness  or  the  tone,  is 
just  a  single  point  upon  the  sense  scale, — no  more  measurable, 
of  itself,  than  is  *  the  highest  point '  of  Mt.  Vesuvius.  The  only 
thing  that  we  can  measure  is  the  distance  between  two  sensa- 
tions or  sense  points,  and  to  do  this  we  must  have  our  unit  step 
or  unit  distance. 

Let  us  take  some  instances.  Suppose  that  two  rooms  of 
equal  dimensions  are  illuminated  by  two  ground  glass  globes, 
the  one  containing  five  and  the  other  two  incandescent  lights  of 
the  same  candle-power.  We  can  say,  by  eye,  that  the  illumina- 
tion of  the  first  room  is  greater  than  that  of  the  second.  How 
much  greater,  we  cannot  possibly  say.  Even  if  the  globes  are 
removed,  so  that  we  can  count  the  lights,  we  cannot  say.  The 
stimuli  stand  to  one  another  in  the  ratio  5  :  2.  But  the  corre- 
sponding sensations  are  simply  different  as  more  and  less,  the  one 
a  '  more  bright '  and  the  other  a  *  less  bright.'  The  brightness 
of  the  lighter  room  does  not  contain  within  it  so  and  so  many 
of  the  brightnesses  of  the  darker  room.  Each  brightness  is  one 
and  indivisible.  What  we  have  given  is  rather  this  :  that  on  the 
scale  of  brightness  intensities,  which  extends  from  the  just  notice- 
able shimmer  of  light  to  the  most  dazzHng  brilliance  that  the  eye 
can  bear,  the  illumination  of  the  one  room  lies  higher,  that  of 
the  other  room  lower.  There  is  a  certain  distance  between  them. 
If  we  can  establish  a  sense  unit  for  this  distance,  we  shall  be 
able  to  say  that  the  greater  brightness  is,  in  sensation,  so  and  so 
many  times  removed  from  the  lesser  brightness  ;  just  precisely 
as  the  top  of  the  mountain,  in  terms  of  the  i  ft.  unit,  is  4200 
times  removed  from  the  bottom.  Neither  of  the  two  bright- 
ness sensations  is  itself  a  magnitude.  The  magnitude  is  the 
distance  which  separates  them  on  the  intensive  brightness 
scale. 

Again  :  we  can  say  by  ear  that  the  roar  of  a  cannon  is  louder, 
very  much  louder,  than  the  crack  of  a  pistol.  But  the  cannon 
roar,  as  heard,  is  not  a  multiple  of  the  pistol  crack,  does  not  con- 


xxvi  InUvdiiction  :  Mental  Measurement 

tain  so  and  so  many  pistol  cracks  within  it.  What  we  have 
given  is  that,  on  the  scale  of  noise  intensities  ranging  from  the 
least  audible  stir  to  the  loudest  possible  crash,  the  cannon  roar 
lies  very  high,  the  pistol  crack  a  good  deal  lower.  Neither 
noise,  in  itself,  is  a  magnitude ;  both  alike  are  points,  positions, 
upon  an  intensive  scale.  The  magnitude  is  the  distance  between 
them.  With  a  sense  unit  of  noise  distance  estabHshed,  we  can 
measure  this  given  distance,  as  before. 

It  was  said  above  that  the  mistaking  of  the  single  sensation 
for  a  magnitude  is  so  natural  as  to  be  almost  inevitable.  We 
are  constantly  confusing  sensations  with  their  stimuh,  with  their 
objects,  with  their  meanings.  Or  rather — since  the  sensation  of 
pyschology  has  no  object  or  meaning — we  are  constantly  confus- 
ing logical  abstraction  with  psychological  analysis ;  we  abstract  a 
certain  aspect  of  an  object  or  meaning,  and  then  treat  this  aspect 
as  if  it  were  a  simple  mental  process,  an  element  in  the  mental 
representation  of  the  object  or  meaning.  What  is  meant  will 
become  clear  at  once,  if  we  take  a  few  instances.  We  do  not 
say,  in  ordinary  conversation,  that  this  visual  sensation  is  lighter 
than  that,  but  that  this  pair  of  gloves  or  this  kind  of  grey  note- 
paper  is  lighter  than  this  other.  We  do  not  say  that  this  com- 
plex of  cutaneous  and  organic  sensations  is  more  intensive  than 
that,  but  that  this  box  or  package  is  heavier  than  this  other.  We 
do  not  even  say,  as  a  rule,  that  this  tonal  quality  is  lower  than 
that,  but  rather  that  this  instrument  is  flat  and  must  be  tuned  up 
to  this  other.  Always  in  what  we  say  there  is  a  reference  to 
the  objects,  to  the  meaning  of  the  conscious  complex.  It  is  not 
grey,  pressure,  tone,  that  we  are  thinking  of ;  but  the  grey  of 
leather  or  paper,  the  pressure  of  the  box,  the  pitch  of  the  violin. 
Now  the  stimuli,  the  physical  processes,  are  magnitudes  or  quan- 
tities. What  is  more  natural,  then,  than  to  say  that  the  cor- 
responding grey  or  pressure  or  tone  is  also  a  magnitude  or  a 
quantity  }  What  is  more  natural  than  to  read  the  character  of 
the  stimuli,  of  the  objects,  into  the  '  sensations '  with  which  cer- 
tain aspects  of  stimulus  or  object  are  correlated  }  At  any  rate, 
this  is  what  Fechner  did.  Fechner  had  an  inkling  of  the  truth ; 
he  knew  that  sense-distances  are  magnitudes,  and  every  now  and 


§   3-     -^^^  Analogy  xxvii 

then  he  seems  to  look  upon  the  single  sensation  as  merely  the 
limiting  point  of  a  distance,  /.  e.,  as  a  position  upon  some  sense 
scale.  But  his  teaching  is  that  the  sensation  as  such  is  a  magni- 
tude. "  In  general,"  he  says,  "  our  measure  of  sensation  amounts 
to  this  :  that  we  divide  every  sensation  into  equal  parts,  i.  e.y 
into  the  equal  increments  out  of  which  it  is  built  up  from  the 
zero  point  of  its-  existence,  and  that  we  regard  the  number  of 
these  equal  parts  as  determined  by  the  number  of  the  cor- 
responding ....  increments  of  stimulus,  ....  just  as  if  the 
increments  of  stimulus  were  the  inches  upon  a  yard-stick."  This 
is  wrong.  No  sensation  is  a  sum  of  sensation-parts  or  of  sense- 
increments  ;  no  sensation  is  a  measurable  magnitude.  Fechner 
has  transferred  to  sensation  a  point  of  view  that  is  right  for  stimu- 
lus, but  that  introspection  refuses  to  recognise  in  psychology. 

§  3.  An  Analogy. — The  passage  which  we  have  just  quoted 
from  Fechner  reads  in  full  as  follows  :  "  Our  measure  of  sensa- 
tion amounts  to  this  :  that  we  divide  every  sensation  into  equal 
parts,  i.  €.,  into  the  equal  increments  out  of  which  it  is  built  up 
from  the  zero  point  of  its  existence,  and  that  we  regard  the  num- 
ber of  these  equal  parts  as  determined  by  the  number  of  the  cor- 
responding variable  increments  of  stimulus  which  are  able  to 
arouse  the  equal  increments  of  sensation,  just  as  if  the  increments 
of  stimulus  were  the  inches  upon  a  yard-stick."  Notice  that 
Fechner  speaks  of  the  variable  increments  of  stimulus  which 
arouse  the  equal  increments  of  sensation.  This  means,  in  our  own 
terminology,  that  equal  sense-distances  do  not  correspond  always 
to  equal  stimulus  magnitudes :  to  obtain  equal  sense-distances, 
under  different  conditions,  we  must  vary  the  magnitude  of  the 
corresponding  stimuli.  The  fact  is  important :  it  is  also  obvious. 
Go  into  a  small  darkened  room,  and  light  a  candle.  There  is  an 
immense  difference  in  the  illumination  of  the  room.  The  physical 
magnitude,  the  photometric  value  of  the  candle,  corresponds  to  a 
very  wide  distance  upon  the  scale  of  subjective  brightness  inten- 
sities. Light  a  second  candle.  There  is  a  difference  in  the 
illumination,  and  a  marked  difference ;  but  it  is  nothing  like  so 
marked  as  the  first  difference.  The  same  physical  magnitude, 
then,  corresponds  now  to  a  lesser  sense-distance.  Light  a  third 
candle,  and  a  fourth,  and  a  fifth.     A  point  will  soon  come  at 


xxviii  hitroductioii :  Mental  Measurement 

which  the  introduction  of  another  candle  makes  hardly  any  ap- 
preciable difference  in  the  illumination.  The  same  physical  mag- 
nitude now  corresponds  to  a  minimal  sense-distance. 

Facts  of  this  sort  recur  in  all  departments  of  sense,  and  it  is 
part  of  the  business  of  quantitative  psychology  to  take  account  of 
them,  and  to  sum  them  up  in  a  numerical  formula.  What  pre- 
cisely the  programme  of  quantitative  psychology  is  in  this  field, 
what  the  facts  are  with  which  it  has  to  deal,  and  how  these  facts 
are  to  be  grouped  under  laws,  we  shall  best  understand  by  help 
of  an  analogy  from  physics.  The  analogy  was  first  suggested 
by  J.  R.  L.  Delboeuf  (i 831-1896),  late  professor  in  the  Univer- 
sity of  Liege, — a  psychologist  -of  great  originality,  to  whom  we 
owe  the  conception  of  mental  measurement  set  forth  in  §  2,1 — and 
has  been  worked  out  in  detail  by  H.  Ebbinghaus  (b.  1850),  pro- 
fessor in  the  University  of  Breslau  and  editor  of  the  Zeitschift 
fur  PsycJiologie  und  PJiysiologie  der  Sinnesorgajie} 

If  a  magnetic  needle  be  suspended  at  the  centre  of  a  circular 
coil  of  wire,  and  an  electric  current  be  sent  through  the  wire,  the 
needle  is  deflected  from  its  position  of  rest.  Suppose  that  we 
iare  seeking  to  discover  the  law  of  this  deflection,  to  find  a  gen- 
eral expression  for  the  movement  of  the  needle  under  the  in- 
fluence of  currents  of  different  strength.  We  send  a  current  of 
so  and  so  many  amperes  through  the  coil,  and  measure  the  arlgle 
of  deflection  upon  a  circular  scale  ;  then  we  send  through  a  current 
of  so  and  so  many  more  amperes,  and  measure  again  ;  and  so  on. 
We  find  that  the  needle  moves  farther  and  farther,  as  the  cur- 
rent is  made  stronger  and  stronger.  But  we  find  also  that  the 
angle  of  deflection  is  not  simply  proportional  to  the  strength  of 
current.  If  we  increase  the  strength  of  current  by  equal 
amounts,  the  deflection  of  the  needle  becomes  progressively 
smaller  and  smaller.  And  however  strong  we  make  our  current, 
the  needle  will  never  make  an  excursion  of  90°.  The  mathe- 
matical expression  of  the  relation  is  very  simple.  If  a  is  the 
number  of  amperes  in  the  current,  k  a  constant,  and  Q  the  angle 
of  deflection,  then  a  =  k  tan.  Q- 

1  Revue  philosophique,  v.,  1878,  53;  Exanien  critique  de  la  loi  psychophysique,  sa 
base  et  sa  signification,  1883,  104  f. 

2  See  this  Zeitschrift,  i.,  1890,  447  ff. 


§  4-      Three  Problems  of  Quantitative  Psychology     xxix 

Let  us  now,  instead  of  taking  determinate  strengths  of  cur- 
rent, change  the  strength  of  the  current  continuously,  and  let  us 
watch  the  behaviour  of  the  needle.  We  find  the  same  law  in 
operation ;  but  it  is  crossed  by  a  second  law.  If  the  needle  is 
hanging  steady,  whether  at  the  zero  point  of  its  scale  or  at  any 
other  point  at  which  it  is  held  by  the  current  in  the  coil,  and  we 
increase  the  current  very  slowly,  we  get  at  first  no  movement  at 
all.  During  this  period,  while  the  needle  remains  stationary, 
our  law  of  correlation  is,  apparently,  not  fulfilled  ;  and  the  greater 
the  increase  of  the  current  before  movement  sets  in,  the  greater, 
of  course,  is  the  apparent  deviation  from  the  law.  Presently, 
however,  when  the  current  has  been  increased  by  a  certain 
amount,  the  needle  goes  with  a  little  jump  to  the  position  which 
the  law  of  correlation  requires.  And  as  we  continue  slowly  to 
increase  the  strength  of  the  current,  the  phenomenon  is  repeated, 
until  the  limit  of  the  needle's  excursion  is  reached.  The  law  of 
correlation  is  not  really  in  abeyance ;  it  is  crossed  or  masked  by 
another  law. 

We  have,  then,  two  things  before  us.  On  the  one  hand,  the 
needle  is  a  magnetic  needle,  and  the  amount  of  its  deflection  is  a 
continuous  function  of  the  current  in  the  coil.  On  the  other 
hand,  the  needle  does  not  move  without  friction  ;  so  that  we  ob- 
tain, under  the  conditions  of  our  second  experiment,  not  a  con- 
tinuous movement  of  the  needle,  but  a  discrete  movement,  a 
series  of  jerks.  It  is  one  and  the  same  needle  that  moves,  and 
one  and  the  same  movement  that  it  makes  ;  but  the  single  needle 
is  at  once  mechanical  and  magnetic,  and  the  single  movement 
gives  evidence  of  the  operation  of  two  distinct  laws. 

§  4.  Three  Problems  of  duantitative  Psychology. — We  cannot 
argue  from  the  behaviour  of  a  magnetic  needle  to  the  behaviour 
of  our  sensations.  Nevertheless,  the  analogy  serves  to  give  us 
our  bearings  in  the  matter  of  sense  measurement.  We  will  take, 
first,  the  facts  of  sensation  that  correspond  to  the  phenomena  of 
friction  in  the  needle. 

(i)  The  perfect  sensation  has  four  attributes:  intensity, 
quality,  duration,  extent.  Every  one  of  these  attributes  of  sen- 
sation is  correlated  with  some  property  of  stimulus.  Not  every 
value  of  stimulus,  however,  is  capable  of  arousing  the  correspond- 


XXX  Introduction:  Mental  Measurement 

ing  sensation.  Just  as  the  current  must  be  increased  by  a  cer- 
tain amount,  whether  from  zero  or  from  some  positive  mag- 
nitude, to  produce  a  deflection  of  the  stationary  needle,  so  must 
the  stimulus  in  every  case  reach  a  certain  magnitude,  if  it  is  to 
set  up  a  sensation. 

Take  intensity.  Some  lights  are  too  faint  to  be  seen ;  there 
are  stars,  e.  g.y  that  even  on  the  darkest  night  remain  invisible 
to  the  naked  eye.  Some  sounds  are  too  faint  to  be  heard ;  we 
may  be  sure,  from  his  gestures,  that  our  friend  is  shouting  to  us, 
but  we  are  too  far  off  to  hear  his  voice.  Some  pressures  are  too 
weak  to  be  sensed ;  we  have  no  knowledge  of  the  flake  of  cigar 
ash  that  falls  upon  our  hand.  Some  tastes  are  too  weak  to  be 
sensed ;  a  draught  of  water  which  proves,  on  chemical  analysis, 
to  hold  various  salts  in  solution,  may  yet  be  entirely  tasteless. 
Some  smells  are  too  weak  to  be  sensed ;  we  get  no  sensation 
from  an  exposure  of  i  mm.  of  black  rubber,  tubing  on  the  olfac- 
tometer. One  of  the  things  that  we  have  to  do,  then,  is  to  deter- 
mine the  least  intensity  of  stimulus,  in  the  different  sense  depart- 
ments, that  will  arouse  a  noticeable  sensation.  What  is  the 
faintest  light  that  we  can  see }  The  least  sound  that  we  can 
hear .?     The  lightest  weight  that  we  can  feel .?     And  so  on. 

Again  :  some  differences  of  stimulus  intensity  are  too  small  to 
be  remarked.  If  a  few  lights  go  out  in  a  brilliantly  lighted  ball- 
room, the  illumination  is  not  sensibly  diminished.  If  an  orches- 
tra is  playing,  and  a  belated  second  violin  suddenly  joins  his  fel- 
lows, the  volume  of  sound  is  not  sensibly  increased.  If  we  lift, 
blindfold,  two  glasses  of  water,  from  one  of  which  a  teaspoonful 
has  been  taken,  we  cannot  say  which  is  the  heavier.  If  we  put 
five  lumps  of  sugar  into  our  coffee,  we  shall  hardly  make  it 
sweeter  by  adding  a  sixth.  If  we  have  spilled  a  bottle  of  eau  de 
Cologne  on  the  carpet,  we  shall  not  notice  the  little  that  our 
guest  has  poured  on  her  handkerchief.  Another  thing  to  do, 
therefore,  is  to  determine  the  least  increase  or  decrease  of  stimu- 
lus intensity,  in  the  different  sense  departments,  that  will  make 
a  noticeable  change  in  the  intensity  of  a  given  sensation,  that 
will  shift  it  a  minimal  distance  up  or  down  upon  the  intensive 
scale. 

The  same  thing  holds  of  quality.     There  is  a  lower  limit  to 


§  4«      TJirce  Problems  of  Quantitative  Psychology    xxxi 

the  tonal  scale  and  to  the  band  of  spectral  colours  ;  the  air  waves 
and  ether  waves  must  reach  a  certain  frequency  of  vibration  be- 
fore they  set  up  the  lowest  audible  tone  and  the  most  extreme 
spectral  red.  Moreover,  although  we  can  distinguish  several 
tones  within  the  musical  semitone,  and  many  more  than  New- 
ton's seven  colours  in  the  spectrum,  still,  not  every  change  of 
vibration  frequency  produces  a  change  of  sense  quality.  We 
cannot  distinguish  an  a^  of  440  vs.  from  one  of  440.1  vs.;  we 
cannot  distinguish    a  blue  of  465/*^  from  one  of  464.5/*/*. 

The  same  thing  holds  of  duration.  If  a  strip  of  spectral  green 
be  exposed,  under  certain  experimental  conditions,  for  a  very 
short  time,  it  is  seen  merely  as  a  grey.  If  the  time  of  exposure 
be  made  a  little  longer,  it  is  seen  as  bluish  grey.  If  the  time  be 
still  further  increased,  the  bluish  grey  becomes  a  distinct  blue. 
Finally,  with  still  greater  increase,  the  green  is  seen  as  green. 
Similarly,  if  the  a^  be  sounded  for  only  ^^  sec,  so  that  a 
single  v.  reaches  the  ear,  we  hear  not  a  tone  but  merely  a  noise. 
As  the  duration  of  stimulus  is  increased,  the  tonal  quality  be- 
comes clear.  If  pairs  of  clicks  are  sounded,  in  rapid  succession, 
a  practised  observer  can  distinguish  a  pair  separated  by  o.  3  sec. . 
from  a  pair  separated  by  0.303  sec.  But  the  same  observer  can- 
not distinguish  a  separation  of  0.3  sec.  from  a  separation  of,  say, 
0.301  sec. 

Finally,  the  same  thing  holds  of  extent.  Two  stars,  whose 
apparent  distance  is  less  than  30'^,  are  always  seen  as  a  single 
star.  Cut  a  square  of  blue  paper,  with  sides  of  i  mm.,  and 
paste  it  on  a  white  card.  Walk  backwards  from  the  card  :  at  a 
distance  of  about  3  m.  the  colour  of  the  blue  disappears.  At 
certain  parts  of  the  skin,  pressures  of  quite  considerable  area  are 
sensed  as  mere  points.  Again  :  a  strip  of  red,  10  cm.  long,  can- 
not be  distinguished  from  a  similar  strip,  10.05  cm.  long,  laid  in 
the  same  straight  line  with  it.  In  pressure  experiments  at  cer- 
tain regions  of  the  skin  it  is  necessary  to  increase  the  diameter 
of  the  stimulus  from  2  to  25  mm.,  in  order  to  obtain  a  noticeable 
difference  of  extent. 

(2)  Further :  our  magnetic  needle  will  never  make  an  excur- 
sion of  90°,  whatever  the  intensity  of  current  that  we  employ. 
In  the  same  way,  the  sense  organs  refuse  to  mediate  sensation 


xxxii  Introduction  :  Mental  Measurement 

when  the  stimulus  has  passed  a  certain  maximal  value.  We  can- 
not hear  'tones'  of    100,000  vs.  in  the   i  sec;  we  cannot  see 

*  colours '  beyond  the  extremest  violet  of  the  spectrum.  Noises  of 
maximal  intensity  stun  or  deafen  us.  If  a  tone  or  pressure  is 
continued  beyond  a  certain  time  limit,  we  cease  to  hear  and  feel ; 
the  organ  gives  out,  becomes  fatigued  and  exhausted.  If  visual 
stimuli  are  presented  at  certain  distances  from  the  fovea,  they  are 
not  seen  ;  the  extent  of  the  field  of  vision  is  Hmited.  Here,  then, 
is  a  second  principal  problem  :  the  determination  of  the  highest 
value  of  stimulus  that  is  still  effective  for  sensation. 

(3)  Thirdly,  we  will  take  the  facts  of  sensation  that  correspond 
to  the  law  of  angular  deflection  in  the  tangent  galvanometer. 
Sensation,  under  all  its  four  aspects,  is  a  continuous  function  of 
stimulus.  True,  a  continuous  gradation  of  stimulus  will,  under 
certain  conditions,  give  us  a  discrete  sense  scale.  But  that  is 
simply  because  the  nervous  structures  involved  offer  resistance 
to  the  incoming  disturbance,  and  so  produce  the  phenomena  of 

*  friction.'  What,  now, — apart  from  friction, — is  the  law  of  cor- 
relation of  stimulus  and  sensation  } 

Let  us  pause,  for  a  moment,  before  we  try  to  answer  this  ques- 
tion, and  look  back  upon  the  path  that  we  have  already  travelled. 
It  should  be  clear  that  we  have  not  as  yet  entered  upon  our 
proper  task  of  mental  measurement.  We  have  talked  of  mea- 
suring the  stimulus  that  can  just  arouse  a  sensation  ;  the  stimu- 
lus that  can  just  evoke  a  change  in  sensation,  /.  ^.,  that  corre- 
sponds to  a  minimal  sense-distance ;  and  the  stimulus  that  can 
still  just  arouse  a  sensation.  All  the  measurements  so  far  dis- 
cussed, therefore,  are  physical,  not  psychological.  Our  present 
problem  requires  us  to  measure  sensation. 

Measurement  demands  three  terms.  Suppose,  then,  that  a 
sense-distance  is  given  :  say,  a  distance  of  sound  intensity.  We 
have  two  sharply  defined  noises,  of  markedly  different  intensity, 
chosen  from  the  middle  portion  of  the  intensive  scale :  we  are  to 
apply  measurement  to  this  noise  distance,  we  are  quantitatively  to 
estimate  it.  How  shall  we  begin }  Well,  we  can  measure  it, 
in  terms  of  an  arbitrary  unit,  simply  by  halving  it.  We  can  ask, 
and  seek  methodically  to  answer  the  question  :  What  noise  inten- 
sity lies  midway,  for  sensation,  between  the  two  given  noises  ? 


§    4-      Three  Problems  of  Quantitative  Psychology  xxxiii 

In  other  words  :  At  what  point,  upon  the  intensive  noise  scale,  is 
the  given  noise  distance  divided  into  two  sensibly  equal  noise  dis- 
tances ?  Having  determined  this  point,  the  middle  noise  inten- 
sity,— which  we  can  do  with  practice,  by  help  of  one  of  the 
metric  methods, — we  may  go  on  to  ask  :  What  point  upon  the 
scale  lies  as  far  above  the  upper  point  of  our  given  distance  as 
this  lies  above  the  middle  point  that  we  have  just  determined  ? 
And  again  :  What  point  lies  as  far  below  the  lower  point  of  the 
given  distance  as  this  lies  below  the  middle  point  ?  These  two 
new  points  established,  we  can  say  that  the  whole  noise  distance 
which  we  have  so  far  explored  is  the  fourfold  of  our  arbitrary 
unit,  the  half  of  the  original  distance.  The  procedure  can  then 
be  continued  above  and  below,  until  a  wide  range  of  noise  inten- 
sities has  been  measured  ;  i.  ^.,  until  a  considerable  section  of  the 
intensive  scale  has  been  marked  off  in  equal  sense  divisions. 


Fig.  I. 

A  diagram  will  simplify  matters.  Let  the  horizontal  line  in 
Fig.  I  represent  the  continuous  scale  of  sensation  intensity  in  the 
sphere  of  noise.  We  have  given  the  two  noises,  the  two  sense 
points,  m  and  o.  Our  first  task  is  to  determine  the  noise  n  that 
lies  midway,  for  sensation,  between  in  and  o.  That  done,  we 
can  take  the  distance  no  as  given,  and  increase  the  intensity  of  a 
till  we  each  a  point  /  such  that  no=op.  Again,  we  can  take  mn 
as  given,  and  decrease  m  till  we  reach  a  point  /,  such  that  lm=mn. 
The  sense-distance  Ip  is  then  four  times  the  sense  unit  lm=^mn= 
no^rz^op.  And  we  can  evidently  go  on  to  determine  q^  r,  .  .  .  , 
and  k^jy...,    in  the  same  manner. 

All  that  we  now  have  to  do,  in  order  to  formulate  our  law  of 
correlation,  is  to  write  the  corresponding  stimulus  values  by  the 
side  of  the  unit  sense-distances.  We  have  already  seen  that 
equal  stimulus  magnitudes  do  not  correspond  to  equal  sense-dis- 
tances, i.  e.,  that  the  law  of  correlation  does  not  take  the  form  of 
a  simple  proportionality  of  the  two.  We  shall  now  discover  what 
other  and  less  simple  relation  obtains. 


xxxiv  Introduction  :  Mental  Measurement 

The  correlation  has  been  worked  out  with  some  degree  of 
fullness  for  brightnesses,  partially  for  noises,  and  a  formula  has 
been  found  which  satisfies  the  results  in  both  cases.  Where, 
however,  the  stimuli  must  be  presented  to  the  observer  success- 
ively (sounds,  lifted  weights,  temperatures,  smells,  etc.)  the  pro- 
cedure is,  naturally  much  more  difficult  and  uncertain  than  it  is 
where  they  can  be  presented  simultaneously  (brightnesses).  By  the 
time  that  the  third  stimulus  comes,  the  observer  may  have  '  for- 
gotten '  the  distance  that  separated  the  second  from  the  first ;  or 
the  presenting  of  the  second  distance  may  itself  drive  the  first 
distance  out  of  clear  consciousness.  Hence  we  are  led  to  look 
round  for  another  method.  Besides,  there  is  the  question  of  the 
unit.  It  is  not  satisfactory  to  start  out  with  an  arbitrary  sense- 
distance,  and  take  the  half  of  it  as  our  provisional  unit.  It 
would  be  far  better  if  we  could  find  a  definite  unit,  to  be  em- 
ployed by  all  investigators  alike.  Is  there  another  method  of 
working,  and  can  we  discover  such  an  unit } 

Suppose  that  a  brightness  or  a  noise  is  given,  knd  that  we  seek 
to  determine  the  just  noticeably  brighter  brightness,  or  the  just 
noticeably  louder  noise.  The  experiment  is  identical  with  one  of 
our  *  friction  '  experiments  ;  and  its  result  is  the  ascertainment 
of  a  just  noticeable  sense-distance.  Let  us  perform  it  at  various 
points  of  the  sense  scale,  so  that  we  get  the  stimulus  values  cor- 
responding to  the  distances  //',  ll'\  ....  mm\  mm" ^  .  .  .  . 
mi\  n'n"y  .  .  etc.  Now  these  are  all  least  distances,  minima  of 
sensible  distance.  Are  they  not,  then,  equal  distances  .?  And, 
if  they  are  equal,  may  we  not  take  them  as  the  units  of  sense 
measurement .? 

That  least  steps,  at  various  parts  of  the  sense  scale,  should  also 
be  equal  steps  is  by  no  means  self-evident.  A  given  difference 
between  sensations  might  be  the  least  perceptible  difference,  and 
yet,  when  compared  with  another  least  perceptible  difference 
from  another  part  of  the  scale,  might  be  larger  or  smaller  than 
this  other.  The  equality  of  just  noticeable  differences  must, 
then,  be  proved  :  it  cannot  be  assumed.  The  appeal  lies,  directly, 
to  introspection  ;  indirectly,  to  the  results  of  experiment.  Fech- 
ner  asserted,  on  the  basis  of  his  own  introspections,  that  all  just 
noticeable  differences  of  sensation  are  sensibly  equal.     The  state- 


§    4-      TJn'cc  Problems  of  Quantitative  Psychology   xxxv 

ment  is  made  positively,  and  Fechner  was  an  exceedingly  care- 
ful and  eminently  practised  experimenter.  Still,  this  evidence 
needs  to  be  supplemented  :  judgments  of  just  noticeable  differ- 
ence are  far  from  easy,  and  one  might  deceive  oneself  in  the 
matter.  Further  evidence  is,  however,  forthcoming.  The  course 
of  stimulus,  over  against  the  least  distances  of  noise  and  bright- 
ness, is  precisely  the  same — is  summed  up  in  the  same  formula 
— as  it  is  over  against  the  larger  sense-distances  with  which  we 
have  been  dealing.  Moreover,  the  formula  which  connects  stim- 
ulus magnitude  with  least  sense  steps  has  the  same  form  in  other 
sense  departments  (pressure,  lifted  weights,  tone,  smell)  that  it 
has  in  the  cases  of  noise  and  brightness.  We  can  hardly  doubt 
then,  that  the  proposition  "All  just  noticeable  differences  of  sen- 
sation are  equal  sense-distances  "  is  correct.  And  so  we  have 
our  new  method  and  our  common  unit  of  sense  measurement. 

We  have  our  unit  of  measurement,  that  is  to  say,  for  intensity 
of  sensation  :  for,  so  far,  we  have  been  speaking  only  of  intensity. 
Does  the  same  thing  hold  of  quality,  duration  and  extent  that 
holds  of  intensity .? 

No :  matters  are  more  complicated.  In  the  case  of  tonal 
pitch,  e.  g.y  it  seems  that  one  and  the  same  formula  would  natur- 
ally (in  bare  sensation)  cover  both  just  noticeable  differences  and 
the  halves  of  larger  tonal  sections  or  distances.  Only,  when  we 
are  asked  to  bisect  a  tonal  distance,  we  are  not  in  the  domain  of 
bare  sensation  ;  we  fall  under  the  influence  of  musical  training 
and  tradition,  and  the  results  of  our  experiments  are  thus  ob- 
scured. Esthetics  has  cut  across  psychology.  Again  :  when 
we  are  determining  the  j.  n.  d.  of  visual  extent,  we  get  one 
formula  for  the  correlation  of  stimulus  and  sensation  ;  when  we 
are  comparing  visual  extents  as  wholes,  we  get  another  and  a 
different  formula.  There  is  good  reason  for  this  result :  the  con- 
ditions of  judgment  are  very  different  in  the  two  cases.  Here  it 
is  psychology  that  cuts  across  psychology  ;  the  one  set  of  experi- 
ments brings  out  one  law,  the  other  set  brings  out  another.  The 
same  thing  seems  to  be  true  of  durations ;.  though  in  this  case 
the  conditions  of  judgment  are  still  more  compUcated,  and  exper- 
iments have  not  been  made  in  sufficient  number  to  allow  of  any 
general  formulation.     The  facts  will  come  out  later,  as  we  per- 


xxxvi  Int7vduction  :  Mental  Measurement 

form  the  experiments.  What  is  important  to  remember  now  is 
that  quantitative  psychology  sets  us  certain  definite  problems  of 
mental  measurement,  and  that  we  are  able  in  theory  to  solve  all 
these  problems — even  if  they  have  not  yet  been  all  solved  in 
fact — by  help  of  our  metric  methods  and  our  unit  of  sense- 
distance. 

There  is  one  point  in  the  above  discussion  which  may  have  puzzled  the 
reader.  We  made  a  sharp  distinction  between  the  phenomena  of  friction 
(the  needle  as  mechanical)  and  the  phenomena  of  correlation  (the  needle  as 
magnetic).  Yet,  in  determining  our  unit  of  mental  measurement,  /.  ^.,  our 
unit  of  correlation,  we  have  had  recourse  to  one  of  the  friction  experiments. 
There  is,  however,  no  real  confusion.  The  fact  that  the  needle  remains 
stationary  for  a  little  while,  as  the  strength  of  current  is  slowly  and  continu- 
ously increased,  is  a  fact  of  friction ;  but  when  the  needle  moves,  it  moves 
in  obedience  to  the  law  of  correlation,  it  moves  as  a  magnetic  needle.  In 
psychology,  we  use  the  fact  of  friction  as  a  means  to  our  end,  which  is 
correlation ;  we  do  not  use  it  for  its  own  sake.  Suppose  that  we  gave  a 
quickly  changing  continuous  stimulus,  like  the  tone  of  a  siren-whistle,  which 
rises  from  bass  to  treble :  we  should  get  a  corresponding  continuous  change 
in  sensation  ;  we  should  hear  what  we  call,  loosely,  a  '  rising  tone.'  How 
could  this  continuously  changing  sense-continuum  help  us  towards  men- 
tal measurement?  How  could  it  give  us  a  correlation  of  stimulus  with 
single  sensation  quality  ?  How  could  it  give  us  the  unit  sense-distance  ?  To 
measure,  we  must  have  a  discrete  series  of  sensations,  in  which  each  term  is 
equidistant  from  its  neighbours  before  and  after.  We  obtain  this  series  by 
increasing  our  stimulus  very  slowly,  so  that  sensation  follows  it,  not  con- 
tinuously, but  in  little  jerks.  We  owe  the  jerkiness  of  sensation  to  the  fact 
of  friction ;  but  the  space  moved  over  at  every  jerk  corresponds  to  a  just 
noticeable  sense-distance,  i.  e.^  if  our  reasoning  is  correct,  to  a  sensation  unit. 

That  there  are  gaps  and  discrepancies,  even  in  this  first  part  of  the  pro- 
gramme of  quantitative  psychology,  ought  not  to  be  surprising.  Think  of 
colours  and  tones.  The  stimulus  magnitudes  progress  uniformly  from  less 
to  greater  in  both  sense  departments.  But  tones  form  one  single  series, 
while  the  colours  correlated  with  homogeneous  light  form  no  less  than  four 
(three  complete,  and  one  incomplete)  series.^  Think,  again,  of  the  differ- 
ence between  a  duradon,  say,  of  0.5  sec,  which  we  can  take  in  as  a  whole, 
as  '  a  '  duration,  and  a  duration  of  5  min.,  which  we  can  estimate  only  by 
means  of  the  number  and  variety  of  the  ideas  and  perceptions  which  '  fill ' 
it.  Here  are  differences  on  the  objective  side.  On  the  subjective,  remem- 
ber that  a  science  does  not  advance  according  to  a  prearranged  logical  plan, 
but  unevenly,  as  the  interests  of  individual  workers  or  the  claims  of  practical 
utility  dictate.      Fechner  was  chiefly  interested   in  the  intensive  aspect  of 

1  See  vol.  i.,  S.  M.,  3,  31  ;  I.  M.,  7  £.  55. 


§   5.  Some   Technical  Terms  xxxvii 

mental  processes,  and  among  mental  processes  in  sensation.  His  example 
has  led  other  enquirers  to  give  a  disproportionate  amount  of  attention  to  the 
laws  of  sensation  intensity.  It  was  a  practical  question,  again,  the  question 
of  the  '  personal  equation  '  in  astronomical  observations,  that  put  reaction 
times  in  the  forefront  of  experimental  psychology.  Further,  at  the  inception 
of  a  science,  problems  will  be  wrongly — or,  at  any  rate,  inadequately — formu- 
lated. Hence  much  of  the  work  done,  e.  g.,  upon  what  were  formerly  called 
the  '  space  sense  '  and  the  '  time  sense  '  is  not  available  for  our  present  pur- 
poses. It  is  concerned  rather  with  perception  than  with  the  sensation  attri- 
butes of  duration  and  extent.  All  these  factors,  objective  and  subjective,  com- 
bine to  make  quantitative  psychology  a  ragged  group  of  facts  and  methods 
rather  than  a  well-rounded  system.     The  rounding  will  come  with  time. 

§  5.  Some  Technical  Terms. — The  magnitude  of  stimulus 
which  corresponds  to  any  just  noticeable  sense-distance  is  called 
a  liminal  stimulus.  The  term  limen  {Schwellcy  threshold)  was 
introduced  into  psychology  by  Herbart  in  181 1  :  a  liminal  stim- 
ulus, or  liminal  stimulus  difference,  is  that  which  lifts  the  sensa- 
tion or  the  sense-difference  over  the  threshold  of  consciousness. 
Our  analogy  of  friction  in  the  galvanometer  indicates  that  all 
liminal  stimuli  do  the  same  sort  of  work,  at  whatever  point  of  the 
sensation  scale  they  are  operating.  The  facts  that  a  stimulus 
must  attain  a  certain  magnitude  in  order  to  arouse  a  sensation  at 
all,  and  that  it  must  attain  a  certain  magnitude  in  order  to  effect 
a  noticeable  change  in  sensation,  are  facts  of  the  same  order. 
Nevertheless,  it  is  customary  (and  it  need  not  be  misleading)  to 
make  a  distinction.  The  magnitude  of  stimulus  that  just  brings 
a  sensation  to  consciousness  is  called  the  stimulus  limen;  the 
magnitude  of  stimulus  that  corresponds  to  a  least  sense  step  from 
some  positive  point  upon  the  sense  scale  we  term,  with  Fechner, 
the  difference  limen. 

The  magnitude  of  stimulus  that  corresponds  to  the  last  notice- 
able sensation  is  called  the  terminal  stimulus :  the  expression 
{Reizhohe)  was  proposed  by  Wundt  in  1 874. 

We  symboHse  '  sensation  '  by  5  or  s,  and  '  stimulus '  by  i?  or  r. 
R  is  the  first  letter  of  the  German  Reiz^  stimulus.  Its  use  is 
not  unnatural  in  English,  since  r  comes  before  s  in  the  alphabet 
as  stimulus  before  sensation  in  the  psychological  experiment. 
*  Stimulus  limen  '  thus  becomes  RL  ;  '  difference  limen  '  DL  ; 
and  *  terminal  stimulus  '  TR. 


xxxviii  Introduction  :  Mental  Measurement 

Note  that  all  these  determinations  may  be  qualitative,  intensive,  temporal 
or  spatial.  Thus  the  qualitative  RL  for  tones  is  an  air  wave  of  some  14 
vs.  in  the  i  sec;  this  magnitude  of  stimulus  gives  the  lowest  audible  tone  ; 
below  it,  there  is  no  tone,  but  either  silence  or  noise.  The  qualitative  DL 
in  the  middle  region  of  the  tonal  scale  is  some  0.25  v.;  two  tones  are  just 
noticeably  different  if  their  pitch  numbers  differ  by  this  amount ;  this  magni- 
tude of  stimulus  corresponds  to  the  least  sense  step,  up  or  down,  from  a  given 
tonal  quality.  The  qualitative  TR  for  tones  is  an  air  wave  of  some  50,000 
vs.  in  the  i  sec;  this  magnitude  of  stimulus  gives  the  highest  audible  tone  ; 
above  it,  there  is  no  tone,  but  either  silence  or  a  hissing  noise.  Similar 
determinations  may  be  made  for  intensity,  time  and  space.  Theoretically, 
the  fourfold  programme  can  be  carried  through  in  all  sense  departments  ; 
practically,  however,  there  are  gaps  in  our  knowledge. 

Note,  further,  that  the  DL  may  be  expressed  in  two  ways,  as  absolute 
and  as  relative.  If  the  ^1  of  44.0  vs.  is  just  noticeably  different  from  a  tone  of 
440.25  vs.,  then  the  absolute  DL  is  0.25  v.,  and  the  relative  DL  %\%  or  -^^^-q. 

§  6.     The  Further  Programme  of  Quantitative  Psychology. — 

We  shall  be  chiefly  occupied,  in  what  follows,  with  the  determina- 
tion of  the  two  limens,  the  RL  and  the  DL^  and  with  the  halving 
of  supraliminal  sense  distances.  These  are,  logically  and  his- 
torically, the  fundamental  things  in  mental  measurement ;  they 
are  the  first  problems  to  be  solved  by  help  of  the  *  psycho- 
physical metric  methods,'  whose  elaboration  we  owe  in  the  first 
instance  to  Fechner.  They  lead  to  a  generalisation,  called  by 
Fechner  *  Weber's  Law,'  in  honour  of  the  physiologist  Ernst 
Heinrich  Weber  (i  795-1 878).  This  law,  roughly  formulated 
by  Weber  in  1834,  on  the  basis  of  experiments  in  touch  and 
sight,  is  the  first  quantitative  uniformity  established  by  psychol- 
ogy. It  has  given  rise  to  a  large  literature,  constructive  and 
controversial,  in  which  all  the  most  important  issues  of  experi- 
mental psychology  have  been  threshed  out,  pro  and  con,  from 
different  points  of  view.  Weber's  Law  is  thus  the  natural  gate- 
way to  quantitative  psychology. 

It  is,  however,  no  more  than  the  gateway.  To  make  Weber's 
Law  and  the  metric  methods  the  be-all  and  end-all  of  quantita- 
tive work  would  be  altogether  misleading.  For  there  is  no 
single  psychological  problem  that  cannot  be  attacked  on  the 
quantitative  as  well  as  on  the  qualitative  side.  Every  one  of  the 
experiments  that  we  have  already  performed,  in  vol.  I.,  can  be 


§   6.  The  Further  Programme  of  Quantitative  Psychology  xxxix 

turned  into  a  quantitative  experiment.  Let  us  take  some  in- 
stances. 

(i)  It  has  been  found  by  experiment  that  the  increase  of 
brightness  which  a  bright  field  gains  by  contrast  upon  a  dark 
ground  is,  within  Hmits,  directly  proportional  to  the  difference 
between  the  brightness  of  field  and  ground.  If  b  is  the  objective 
brightness  of  the  field,  B  that  of  the  ground,  and  c  the  increase 
of  brightness  that  b  gains  by  contrast,  then  c=^k  {b — B)\  where 
>^  is  a  value  that,  under  the  most  favourable  circumstances,  may 
be  ==i.  This  formula  yields  interesting  deductions,  which  can 
be  experimentally  verified ;  and  similar  formulae  may  be  written 
for  colour  contrasts. 

(2)  Of  all  mental  processes,  the  affective  are  the  most 
baffling  and  the  least  amenable  to  experiment.  Neither  of  the 
two  affective  methods  now  in  use — the  methods  of  impression 
and  of  expression^ — is  able,  as  things  are,  to  furnish  quantitative 
results.  Nevertheless,  an  attempt  has  been  made,  even  in  this 
obscure  department  of  psychology,  to  formulate  in  mathematical 
terms  the  law  that  relates  affective  process  to  affective  stimulus. 
It  is  well  known  that  the  greater  the  preexistent  R  the  more 
must  it  be  increased  if  the  increase  is  to  produce  an  appreciable 
change  in  the  corresponding  affection.  When  one  is  thoroughly 
tired  out,  a  little  more  exertion  seems  to  make  no  difference ;  it 
is  the  oncoming  of  tiredness  that  is  distinctly  unpleasant.  When 
one  is  in  severe  pain,  one  can  easily  *  stand  a  little  more ' ;  it  is 
the  lesser  pains  that  show  clear  degrees  of  unpleasantness. 
When  one  is  in  full  health  and  vigour,  the  good  things  of  life  are 
taken  as  a  matter  of  course  ;  it  is  in  convalescence  that  health- 
increments  are  more  and  more  pleasant.  The  simplest  expres- 
sion of  this  law  is  given  by  the  formula  A=^k  log  R  +  log  r, 
where  A  and  R  stand  respectively  for  affection  and  stimulus, 
and  >^  and  c  are  constants. 

This  law  holds  only  under  certain  theoretical  conditions ;  in 
practice,  it  is  cut  across  by  many  other  laws  of  the  affective  life. 
It  would  be  foohsh,  therefore,  to  give  it  any  absolute  value  ;  but 
it  would  be  equally  foolish  to  ignore  the  amount  of  truth — and 
the  hint  of  affective  measurement — which  it  contains. 

1  See  vol.  I,  S.  M.,  92  ff. ;  I.  M.,  149  ff. 


xl  Intro dttction :  Mental  Measuremejit 

(3)  It  seems  a  little  paradoxical,  at  first  sight,  to  say  that  we 
can  measure  our  illusions  :  for  will  not  the  same  influences  that 
distort  perception  baffle  us  in  any  attempt  we  may  make  to  esti- 
mate the  amount  of  distortion  ?  Yet  the  problem  becomes  sim- 
ple enough  if  only  we  regulate  its  conditions.  We  can,  e.  g.y 
measure  an  extent  which,  under  the  conditions  of  illusory  percep- 
tion, is  overestimated,  by  comparing  it  with  a  similar  extent  seen 
under  normal  conditions.  Let  a  be  the  objective  value  of  the 
first  extent,  a'  its  illusory  value,  and  b  the  objective  value  of  the 
second  extent  which  is  =  a'.  Then  b  —  a  gives  us  an  objective 
measure  of  the  amount  by  which  ^,  under  the  conditions  of  the 
illusion,  is  overestimated.  Experimental  psychology  has  been 
busy  of  late  with  this  sort  of  quantitative  estimation  of  the 
geometrical  optical  illusions,  and  the  results  are  of  great  im- 
portance for  the  theory  of  space  perception. 

(4)  Finally,  we  may  refer  to  certain  investigations  into  the 
laws  of  memory.  We  know,  from  everyday  experience,  that  the 
longer  the  time  which  has  elapsed  since  a  given  event  occurred, 
the  more  uncertain  and  inexact  is  our  recollection  of  it.  Experi- 
ence does  not,  however,  tell  us  anything  very  definite  of  the 
nature  of  the  memory  curve.  We  may  have  noticed,  under  ex- 
ceptionally favourable  circumstances,  that  memory  weakens  at 
first  quickly,  and  then  more  slowly ;  but  we  shall  not  have  got 
beyond  this  observation.  Experiment  furnishes  us  with  the  for- 
mula 7=71 — TT-  .*  where  m  stands  for  memory  (amount  retained), 
y  for  forgetfulness  (amount  lost),  t  for  time  elapsed,  and  k  and 
c  are  constants. 

It  is  needless  to  extend  this  list.  Enough  has  been  said  to 
show  that  psychology  may  be  treated,  from  beginning  to  end,  as 
a  quantitative  science.  True,  very  little  has  been  done — we 
might  go  farther,  and  say  that  very  little  has  even  been  attempted 
— as  compared  with  what  remains  to  do.  But  in  principle  every 
single  problem  that  can  be  set  in  psychology  can  be  set  in  quan- 
titative form. 

§  7.  Questions. — ( i )   Write  the  equation  P  =—p  in  its  four 

possible  forms.     Are  all  the  forms  employed  in  scientific  meas- 
urements }     Give  instances. 


§   7-   Questions  xli 

(2)  Define  Psychophysics.  Describe  briefly  the  historical  con- 
ditions under  which  the  science  arose. 

(3)  Discuss  the  statement:  "Sensation,  under  all  its  four 
aspects,  is  a  continuous  function  of  stimulus." 

(4)  Discuss  the  statement  that  least  sense  steps  need  not, 
logically,  be  equal  sense  steps.     Illustrate. 

(5)  Throw  into  quantitative  form  all  the  experiments  worked 
out  qualitatively  in  vol.  I.  State  which  of  these  quantitative 
problems  you  deem  to  be  the  most  important,  and  why. 

(6)  If  two  different  brightnesses  are  given,  and  we  are  re- 
quired to  find  a  third  brightness  that  lies  for  sensation  midway 
between  them,  we  are  said  to  equate  two  sense-distances.  The 
distance,  not  the  sensation,  is  the  magnitude.  How  does  the 
distance  come  to  consciousness }  What  is  the  material  of  judg- 
ment in  the  experiment .? 

(7)  Rewrite  the  affective  formula  of  §  6  (2),  in  order  to  make 
it  square  with  the  hypothesis  that  mental  distances,  not  mental 
processes,  are  measurable. 

(8)  Discuss  the  hypothesis  that  the  RL  and  the  DL  are  facts 
of  the  same  order,  phenomena  of  *  friction.' 

(9)  Go  over  the  list  of  quantitative  experiments  that  you  have 
made  for  Question  (5),  and  separate  those  in  which  the  measure- 
ment is  mental  from  those  in  which  it  is  physical.  What  is  the 
relative  importance,  for  psychology,  of  the  two  classes  of  ex- 
periments ? 


CHAPTER  I 


Preliminary   Experiments 


EXPERIMENT   I 


§  8.  The  dualitative  RL  for  Tones :  the  Lowest  Audible  Tone. — 

The  object  of  this  experiment  is,  on  the  quantitative  side,  pre- 
cisely to  determine  the  lower  limit  of  tonal  hearing  and,  on  the 
quaUtative,  to  gain  introspective  familiarity  with  the  lowest  au- 
dible tones. 

Materials. — Appunn's  lamella.  Scraps  of 
felt  or  baize.  [The  Appunn  lamella  (Fig.  2) 
is  a  blade  or  strip  of  soft  steel,  385  mm.  long, 
1 5  mm.  wide,  and  i  mm.  thick.  It  is  clamped 
in  a  wooden  vise,  which  is  in  turn  to  be  screwed 
to  the  edge  of  a  table.  Along  one  face  is 
marked  a  scale,  which  shows  the  point  at 
which  the  strip  must  be  fixed  in  the  vise  if  it  is 
to  make  4,  5,  6,  .  .  .  .  24  pendular  vibrations 
in  the  i  sec.  The  upper  end  of  the  lamella  is 
riveted  to  a  thin  steel  disc,  about  40  mm.  in 
diameter.  A  cloth  ring,  35  mm.  in  breadth, 
slides  up  and  down  the  strip :  the  purpose  of 
this  ring  is  to  eliminate  possible  overtones,  and 
it  is  best  set,  in  every  experiment,  at  about  one- 
third  of  the  distance  from  the  upper  end  of 
the  strip  to  the  scale  mark  at  which  it  is 
clamped.] 

Adjustment  of  Apparatus. — The  deepest 
tones  are  intrinsically  weak ;  and  an  unprac- 
tised O  finds  it  difficult    to  distinguish  them 
from  noise.     Every  precaution  should  therefore 
Fig.  2.  be  taken  to  make  the  conditions  of  observation 

as  favourable  as  possible.  The  vise  must  be  clamped  firmly  to 
the  edge  of  the  table :  scraps  of  felt  may  be  used  to  prevent 


2  Preliminary  Experiments 

jarring  or  rattling.  The  table  itself  must  be  higher  than  usual, 
and  must  stand  solidly  on  the  floor  :  pieces  of  felt  or  baize  should 
be  placed  under  the  legs. 

(9's  chair  is  so  disposed  that  his  better  ear  is  directly  opposite 
the  source  of  sound  :  i.  e.,  the  lamella  plays  directly  into  (not 
past)  the  opening  of  the  external  meatus.  The  distance  from 
the  lamella,  in  the  vertical  position  of  rest,  to  the  orifice  of  the 
ear  should  be  measured,  and  kept  as  nearly  constant  as  may  be 
throughout  the  experiment.  Since  the  tones  are  weak,  this  dis- 
tance should  be  very  short.  A  chin-rest  may  be  used,  to  ensure 
a  constant  position  of  the  head.  The  orifice  of  the  ear  should 
be  4  or  5  cm.  above  the  upper  horizontal  surface  of  the  vise. 

Whether  or  not  the  unused  ear  shall  be  stopped  depends  upon 
circumstances.  If  the  stopper  (which  may  be  a  plug  of  cotton 
wool  and  laboratory  wax,  or  a  small  cork  softened  in  vaseline) 
render  O  at  all  uncomfortable,  or  serve  in  the  least  degree  to 
distract  his  attention,  it  is  best  to  leave  both  ears  open.  The 
point  should  be  settled  in  the  preliminary  experiments. 

Preliminary  Experiments. — O  must  have  preliminary  prac- 
tice in  the  hearing  of  deepest  tones.  This  is  necessary,  in  order 
that  he  may  acquire  a  standard  of  judgment.  If  we  are  to  de- 
termine a  valid  RL^  every  R  must  be  judged  in  the  same  way, 
from  the  same  point  of  view,  by  the  same  criterion,  as  every 
other.  But  if  6>'s  preliminary  practice  is  insufficient,  he  may 
shift  his  standard  of  judgment  as  the  experiment  proceeds :  he 
may  presently  find  a  tonal  element  in  an  R  which  he  had  formerly 
taken  to  be  a  whirring  or  puffing  noise  ;  or  he  may,  contrariwise, 
come  to  demand  greater  smoothness  and  volume  in  the  R  which 
he  reports  as  tonal.  E  must  also  have  preliminary  practice  in 
the  manipulation  of  the  lamella.  The  instrument  is  awkward,  at 
best ;  and  the  experiment  will  not  run  smoothly  until  he  has 
learned  to  adjust  and  actuate  it  by  a  set  of  rapid  and  accurate 
movements,  without  effort  or  bungling. 

The  limits  of  the  instrument  suggest  that  the  RL  for  tones 
lies  above  4  and  below  24  vs.  in  the  i  sec.  Now  it  is  a  good 
deal  easier,  under  the  conditions  of  the  experiment,  to  say  that 
one  hears  a  tone  than  to  say  that  one  does  not.  The  quickest 
vibrations  of  the  lamella  give  a  distinct  and  unmistakable  tonal 


§  8.     The   Qualitative  RL  for   Tones  3 

quality  ;  the  slower  vibrations  give  a  deep  whir  or  whiz,  which  an 
unpractised  O  may  well  mistake  for  a  very  low  tone.  Hence  it 
will  be  advisable  to  begin  (9's  practice  with  the  highest  available 
tone  of  24  vs.  By  working  downwards,  from  this  upper  limit,  E 
will  be  able  to  establish,  in  a  rough  and  tentative  way,  the  posi- 
tion of  0\  RL.  This  preliminary  knowledge  is  essential  for  the 
further  course  of  the  experiment. 

Suppose,  now,  that  (9's  left  ear  is  to  be  tested.  The  lamella  is 
clamped  to  the  edge  of  a  high,  solid  table.  The  chair  (with  its 
chin-rest)  is  brought  up  to  this  edge.  O  seats  himself  in  such 
a  position  that  the  set-screw  of  the  vise  Hes  just  above  and  in 
front  of  his  left  shoulder.  E^  standing  behind  O,  actuates  the 
lamella  by  laying  the  tip  of  his  left  thumb  over  the  upper  edge 
and  nearer  surface  of  the  disc,  and  pulling  out  disc  and  strip 
to  his  left.  The  amount  of  pull  is  determined  by  the  stiffness 
and  elasticity  of  the  lamella ;  no  definite  direction  can  be  given 
with  regard  to  it.  He  then  releases  the  disc,  by  a  quick  and 
clean  lift  of  the  thumb  ;  there  must  be  no  catching  or  dragging. 
To  readjust  the  lamella,  he  turns  the  set-screw  (over  6>'s  shoulder) 
with  his  right  hand,  and  raises  or  lowers  the  lamella  with  his 
left.  It  may  be  necessary  for  O  to  swing  out  a  little,  to  his 
right,  while  the  adjustment  is  being  made  :  but  this  is  not  always 
the  case.  If  the  lamella  becomes  bent,  by  a  too  violent  pull,  it 
must  be  removed  from  the  vise  and  straightened  before  the  experi- 
ment is  continued. 

O  is  to  know  nothing  of  the  result,  and  as  little  as  possible  of 
the  course,  of  these  preliminary  experiments.  His  eyes  are  closed 
throughout.  He  knows  that  E  is  working  from  above  down- 
wards :  but  that  is  all.  E  may  sound  any  particular  tone  (that 
of  24,  or  22,  or  18,  etc.,  vs.)  half-a-dozen  times  over,  or  may  sound 
it  once  only.  He  must  make  the  motions  of  readjustment  at 
every  step,  but  he  need  not  actually  readjust.  Hence  O  cannot 
guess  at  the  value  of  R  which  corresponds  to  his  RL, 

Method. — E  knows,  in  a  general  way,  the  position  of  (9's 
RL.  He  is  now  to  determine  its  position  exactly.  To  this  end, 
he  prepares  a  set  of  20  blank  forms,  each  outlining  an  experimental 
'series.'  Ten  of  these  series  are  ' descending '  ;  in  them,  E 
works  down  to  the  RL  from   above,  from  a  distinct  tone.     Ten 


4  Preliminary  Experiments 

are  '  ascending ' ;  in  them,  E  works  up  to  the  RL  from  below^ 
from  a  distinctly  toneless  noise.  In  the  actual  experiment,  each 
series  is  gone  through  separately.  O,  that  is  to  say,  is  first  sub- 
jected, in  regular  order,  to  the  7?  of  a  descending  series,  and  says 
at  every  step  whether  or  not  he  hears  a  tone.  Then,  after  a 
pause,  he  is  subjected  in  the  same  way  to  the  R  of  an  ascending 
series,  and  again  says  at  every  step  whether  or  not  he  hears  a 
tone.  This  procedure  is  continued,  until  all  20  series  have  been 
applied. 

We  must  now  look  at  the  Method  in  a  little  more  detail. 

(i)  Cofistructioii  of  the  series. — E  found,  in  his  preliminary  experiments, 
that  (9's  RL  lies  in  the  neighbourhood  of  a  certain  R  :  say,  an  i?  of  ;r  vs. 
The  blank  form  of  a  descending  series  may  then  contain  the  y?-values  ;f  +  4, 
X  -\-  ;i,  X  -h  2,  X  +  i-i  Xj-  and  the  blank  form  of  an  ascending  series  the  up- 
values X  —  4,  ;r  —  3,  0-  —  2,  X —  I,  X.  In  the  actual  experiment,  O  will 
first  be  given  the  7?-values  ;ir  +  4,  ;r  4-  3,  .  .  .  .  and  then,  after  a  pause, 
the  7?-values  x — 4,  ;r — 3,  .  .  .  .  and  will  inform  E  at  every  step 
whether  or  not  he  hears  a  tone. 

(2)  Size  of  the  steps  in  the  series. — It  is  clear  that  our  steps  must  be  as 
small  as  we  can  conveniently  make  them.  We  are  to  determine  the  RLj 
and  the  smaller  our  steps,  the  more  closely  (other  things  equal)  shall  we 
approximate  to  its  real  value.  In  the  present  instance,  we  may,  for  sim- 
plicity's sake,  take  the  unit  of  the  lamella  (i  v.)  as  the  fixed  step  for  all 
series  alike. 

(3)  Startingpoint  of  the  series. — It  is  unwise  to  begin  every  descending 
and  every  ascending  series  with  the  same  ^-value.  Thus,  it  would  be  un- 
wise to  begin  every  descending  series  with  ;r  -f-  4,  and  every  ascending 
series  with  x  —  4  vs.  If  we  do  this,  O  will  soon  become  aware  that  all 
the  \  and  all  the  f  series  are  alike.  The  series  are  so  short,  that  he  will 
be  able  to  remember,  as  each  fresh  series  comes,  how  he  judged  in  preced- 
ing series.  We  want  him  to  pay  strict  and  undivided  attention  to  every  7?, 
as  it  is  presented  to  him.  Instead  of  that,  he  will  very  probably  be  think- 
ing :  "  This  is  the  fourth  step  :  I  heard  a  tone  at  that  step  before, — so  that 
I  ought  to  hear  a  tone  now  :  yes  !  I  hear  a  tone  ! " 

If,  then,  we  begin  one  ^  series  with  jir  +  4,  we  should  begin  another  with 
;f  -H  8,  etc.  And  if  we  begin  one  f  series  with  x  —  4,  we  should  begin 
another  with  x  —  8,  etc.  We  may,  indeed,  begin  where  we  like,  provided 
that  we  obey  three  rules.  The  first  is,  that  the  series  shall  not  be  so  long 
as  to  tire  O.  The  second  is,  that  the  starting-point  of  the  series  shall 
always  be  a  clear  noise  or  a  clear  tone :  we  may  not  begin  with  a  doubtful 
R.  And  the  third  is,  that  there  shall  be  (approximately)  as  many  long  and 
short  4,  series  as  there  are  long  and  short  f  series  in  the  total  20.  If  one 
of  the  4.  series  have  4  steps,  one  of  the  \   should  have  4  (or  3  or  5)  steps ; 


§  8.      The  Qualitative  RL  for  Tones  '5 

if  one  of  the  4'  have  lo  steps,  one  of  the  \  should  have  io(or  9  or  1 1)  steps. 
It  may  not  be  possible  to  make  the  lengths  of  the  4'  and  f  series  strictly- 
identical  ;  but  they  must  be  as  nearly  alike  as  we  can  make  them. 

Within  these  rules,  we  may  begin  where  we  like.  Shall  we  begin  at  a  dif- 
ferent point  in  every  series.?  Must  all  our  10  \  series,  e.  g.^  be  of  different 
lengths .?  This  is  not  necessary.  It  may  be  better  for  O's  attention  to  have 
three  sorts  of  series — long,  intermediate,  short — than  to  have  10  different 
series  that  slide  into  one  another  by  imperceptible  degrees.  A  good  deal 
depends,  too,  upon  the  position  of  the  RL.  It  may  be  not  only  inadvisable, 
but  actually  impossible,  to  begin  a  \  series  at  10  different  7?-values.  In  this 
matter  E  must  use  his  own  judgment. 

(4)  The  blank  forms. — It  should  now  be  clear  that  E  cannot  fill  out  all 
his  blank  forms  beforehand.  He  makes  out  his  first  4,  series,  we  will  say, 
as  X  +  4,  ;i'  +  3,  :r  4-  2,  ;ir  H-  I,  ;ir.-  and  he  does  this  on  the  assumption 
(warranted  by  the  preliminary  experiments)  that  O's  judgment  will  change 
from  '  tone  '  to  'no  tone  '  at  x.  But  what  if,  under  the  new  conditions  of 
the  experiment,  (9's  judgment  change  at  ;r  H-  2  ?  Or  what  if  it  do  not 
change  till  x  —  3  ?  In  such  an  event,  the  corresponding  \  series  would 
begin,  not  at  ;ir  —  4,  but  rather  at  x  ox  2X  x  —  10.  And  such  an  event  is 
by  no  means  impossible.  In  view  of  this  uncertainty,  it  will  be  best  to 
enter,  on  all  the  20  blanks,  all  the  possible  i?-values  from  4  to  24.  The 
blanks  may  then  be  numbered  in  the  order  i  to  20 ;  the  direction  of  the 
series  may  be  shown  by  the  sign  ^^  or  >j>  in  the  left-hand  margin  of  the 
blank :  the  points  at  which  each  series  begins  and  ends  will  be  indicated  by 
the  introspections  set  down  opposite  the  7?-values  employed  in  the  series. 
The  starting-point  of  a  new  series  may  then  be  chosen  in  the  light  of  the 
preceding  results. 

(5)  Order  of  the  series. — We  have  taken  it  for  granted  that  the  whole 
experiment  of  20  series  is  to  begin  with  a  \  series,  and  that  the  4-  and  f  series 
are  to  alternate,  regularly,  throughout  the  experiment.  We  begin  with  the 
\  series,  because  O  is  a  comparatively  unpractised  observer,  and  his  first 
series  should  therefore  be  made  as  easy  as  possible.  His  practice  will, 
however,  be  steadily  increasing  as  the  experiment  advances ;  and  we  must 
accordingly  arrange  the  later  series  so  that  the  benefits  of  this  increasing 
practice  shall  be  shared,  as  equally  as  may  be,  between  the  \  and  the  \ 
series.  This  end  may  be  reached  in  various  ways.  Thus,  if  we  divide  the 
20  series  into  four  groups  of  five  each,  the  first  group  might  contain  the 
series  |f  |f  I,  the  second  f  J'f  If ;  and  the  third  might  repeat  the  second, 
and  the  fourth  the  first.  E  should  work  out  other  arrangements  for  him- 
self. 

The  relative  length  of  successive  series  is  best  determined  by  chance. 
If  E  decides  to  use  three  lengths — long,  moderate,  short — he  should  write 
these  words  (6  '  long,'  8  '  moderate,'  6  '  short' )  on  20  cardboard  tickets  ; 
shake  the  tickets  in  a  bag ;  and  draw  a  ticket  for  each  new  series.  If  he 
uses  four   lengths — long,   moderately   long,    moderately    short,    short — he 


6  Preliminary  Experiments 

writes  their  titles  (5  apiece)  on  20  similar  cards,  and  draws  as  before.  And 
so  on. 

(6)  Instructions  to  O. — We  want  O  to  be  impartial  and  attentive.  It  would 
be  best,  from  this  point  of  view,  that  he  should  know  absolutely  nothing  of 
the  method  or  of  the  instrument.  As  a  matter  of  fact,  he  knows  {a)  that  the 
i?-values  employed  cannot  fall  below  4  or  rise  above  24  vs.  He  knows  also 
(J?)  that  the  unit  of  the  series  is  i  v.  He  knows  {c)  that  the  first  series  will 
be  4'.  He  does  not  know  what  the  direction  of  every  subsequent  series  is 
to  be ;  there  are  many  patterns  of  arrangement,  any  one  of  which  E  may 
adopt.  At  the  same  time,  every  series  is  to  begin  either  with  a  clear  tone 
or  with  a  clear  noise.  If  it  begin  with  a  tone,  it  must  be  <i<;  if  it  begin  with 
a  noise,  \.  Since  we  cannot  prevent  (9's  finding  out,  in  this  way,  what  the 
particular  series  is  to  be,  and  since  his  thinking  about  the  matter  might 
possibly  distract  or  fatigue  his  attention,  we  will  make  a  virtue  of  necessity 
and  {d)  tell  him  the  direction  of  each  series  before  we  enter  upon  it. 

On  the  other  hand,  O  knows  nothing  (and  is  to  know  nothing)  of  the 
length  of  the  series.  The  series  now  beginning  may  be  three  steps  or  ten 
steps  long.  O  does  not  know  ;  and,  not  knowing,  must  pay  strict  attention 
to  each  R  as  it  is  presented. 

It  need  hardly  be  said  that  O  is  to  know  nothing,  either,  of  the  nu- 
merical results  of  the  experiment  until  the  whole  20  series  have  been  com- 
pleted. 

All  these  rules  and  directions  may  be  summed  up  in  a  single  sentence. 
E  is  to  use  short,  small-stepped  series,  10  \  and  Jo  \ ,  and  is  to  keep  O  on 
the  alert  by  varying  the  length  of  the  series  within  the  limits  of  fatigue. 

Experiment. — O  is  seated  in  position  :  his  eyes  closed,  and 
his  unused  ear  stopped  or  open,  as  the  case  may  be.  E  has  the 
blank  form  of  his  first  \  series  before  him.  E  first  says 
"  Ready  !  "  as  a  signal  to  O  that  a  series  is  about  to  begin.  He 
then  says  *  Now ! '  and  after  a  short  2  sec.  gives  the  stimulus. 
O  says  *  Tone ! '  and  E  enters  a  ■\-  sign  opposite  the  corre- 
sponding ^-value  in  his  blank  form.  E  then  says  *  Now  ! '  and 
after  a  short    2    sec.  gives   the  next   lower  R.     O  again   says 

*  Tone ! '  and  E  enters  another  -\-  in  the  record.  The  series 
is  continued  in  this  way,  in  a  steady  and  even  rhythm,  until  an 
R  is  given  to  which  O  replies  either  '  Doubtful  ! '  or  '  No  tone  !  * 
E  enters  a    .?    or  - —  in  the  record,  and  the  series  stops. 

A  pause,  perhaps  of  3  min.,  is  now  made.     Then  a  second 

*  Ready  ! '  is  given,  and  a  new  series  begins.  E  starts  from  an 
R  which  evokes  a  decided  *  No  tone  ! '  and  continues  the  R  up 
to  the  point  at  which   O  first  says,  definitely,   '  Tone  ! '     There 


§   8.      The  Qualitative  RL  for  Tones  7 

the  series  stops.     The  record  may  have  the  form  — ,  — ,  — ,  — , 
-(-  :  or  possibly  — ,  — ,  — ,  ?,   -|-  :    etc. 

When  5  series  have  been  taken,  E  and  O  change  places.  After 
another  5  series,  they  change  places  again ;  and  so  on,  until  the 
20  series  have  been  completed  for  each. 

On  no  account  should  the  preliminary  experiments  be  made  on  the  same 
day  as  the  experiment  proper.  The  best  plan  is  to  spend  one  day  upon 
the  preliminary  experiments  and  the  method,  and  to  work  through  the 
regular  series  (after  a  few  more  preliminary  tests)  at  the  next  laboratory 
session.  If  the  work  proves  very  tiring,  three  days  may  be  given  to  it : 
the  experiment  proper  is  then  performed  (each  time  with  a  few  prelimi- 
naries) half  on  the  second  and  half  on  the  third  day. 

O  will  naturally  employ  his  better  ear  for  this  experiment,  whether  the 
unused  ear  be  closed  or  open.  If  the  Instructor  think  it  worth  while,  the 
whole  experiment  may  be  repeated  for  the  other  ear. 

Results. —  E  has  his  20  forms,  which  show  the  ^-values  em- 
ployed in  the  various  series  and  (9's  introspections.  The  forms 
should  be  pasted,  or  the  full  series  copied,  into  the  note-book. 

These  20  records  represent  10* paired'  series:  that  is,  10 
\  and  10  (correspondingly  long)  \  series.  The  \  series  end 
with  the  7?-value  at  which  the  tone  has  become  just  unnoticeable 
(first  judgment  of  t  or  — ).  The  \  series  end  with  the  i? -value 
at  which  the  tone  has  become  just  noticeable  (first  -{-  -  judg- 
ment). Now  the  RL  stands  on  the  dividing  line  between  notice- 
ableness  and  unnoticeableness.  Hence  we  may  determine  it  from 
any  one  of  our  paired  series  by  taking  the  average  of  the  final 
terms  of  the  separate  (|  and  \)  series.  But  we  have  secured 
10  paired  series ;  and  we  have  been  at  this  trouble  because  there 
is  safety  in  numbers :  the  result  of  a  single  paired  series  might 
be  affected  by  all  manner  of  accidents, — an  uncomfortable  posi- 
tion on  (9's  part,  some  outside  noise,  some  temporary  lapse  of 
attention,  etc.  We  proceed,  accordingly,  to  determine  the  aver- 
age value  of  all  20  final  terms, — of  the  10  'just  noticeables' 
and  the  10  'just  unnoticeables.'  This  average  value  fairly 
represents  the  RL  for  the  particular  O,  under  the  particular  con- 
ditions of  experimentation,  and  of  health,  practice,  attention,  etc. 

It  is,  however,  not  enough,  when  we  have  measured,  to  give 
simply  the  average  result  of  our  measurements.  The  results  of 
the  individual  measurements  are  not  identical ;  all  sorts  of  acci- 


8  Preliminary  Experimejits 

dental  circumstances  have  affected  the  separate  tests ;  the  indi- 
vidual results  vary  through  a  certain  range  of  values,  above  and 
below  the  average.  In  the  present  case,  the  20  values  which  we 
have  averaged  are  not  identical ;  and  the  final  RL  need  not  be 
identical  with  any  one  of  them.  This  fact  of  variation  must  find 
quantitative  expression  :  and  it  is  expressed  in  what  is  termed 
the  *  mean  variation  *  or  *  average  deviation '  of  the  average  re- 
sult. The  MV  OY  AD  i^  the  average  amount  by  which  the  sep- 
arate measurement-values  differ  from  their  average.  In  the 
present  case,  then,  it  is  the  average  amount  by  which  the  sepa- 
rate *just  noticeables  '  and  'just  unnoticeables '  differ  from  the 
average  RL.  If  we  represent  these  separate  values  by  the  sym- 
bols RLi^  RL2,  ....  RL2Q,  it  is  the  value  determined  by  the 
formula : 

MV  or  jj)^(^^^-^^^)+(^L^^-J^L)+  '  •  •  -^(J^L2,-RL) 

20 

The  differences  in  this  formula  are  all  to  be  considered  as  positive  differ- 
ences, /.  e.,  are  to  be  summed  up  without  regard  to  sign.  A  divergence 
from  the  average  is  a  real,  positive  divergence,  whether  it  fall  upon  ih.t  plus 
or  minus  side  of  the  average.  Suppose  (for  the  sake  of  an  illustration)  that 
the  average  value  of  the  RL  is  18  vs.;  and  suppose  that  the  values  16  and 
20  occur  among  the  separate  determinations.  In  each  case,  the  difference 
counts  as  a  positive  '  two.'  The  formula  writes  16  —  18  and  20 —  18  ;  but 
the  mimis-s\gn  is  simply  the  sign  of  difference. 

In  general  terms,  the  formula  may  be  written  thus : 

where  A  is  the  average  value  of  the  magnitude  measured  (here  the  RL)  ; 
a^  b^  .  .  .  n  are  the  values  of  the  successive  separate  determinations  (here 
the  separate  'just  noticeables  '  and  '  just  unnoticeables ')  ;  and  A^  is  the 
number  of  these  determinations  (here  20,  the  number  of  series  taken). 

The  mode  of  calculation  may  be  illustrated  as  follows.  Suppose  (the 
numbers  are  merely  fanciful)  that  we  have,  to  calculate  the  value  of  the 
i?Z,  the  six  determinations:  22,  20;  21,  22  ;  19,  18.  The  RL  is  the  arith- 
metical mean  of  these  values.  Summing  them  up,  and  dividing  by  6  (the 
number  of  determinations),  we  find  RL=2o.2,.  Notice  that  there  is  no  actual 
determination  of  this  value.  Now  we  have  to  find  the  MV^  the  average 
difference  between  the  separate  determinations  and  the  RL  of  20.3.  The 
separate  differences  are  1.7,  0.3;  0.7,  1.7;  1.3,  2.3.  Summing  them  up, 
and  dividing  the  sum  by  6,  we  find  their  arithmetical  mean  (the  MV)  to  be 
1.3.  This  value  does  actually  coincide  with  one — but  with  one  only — of 
the  separate  differences.     The  final  result  of  our  measurements  is  then  stated 


§   8.      TJie   Qualitative  RL  for    Tones  9 

as  20.3  _[-  1.3.  This  means  that,  under  the  conditions  of  the  experiment, 
the  dividing  line  between  noticeableness  and  unnoticeableness  is  drawn  for 
O^  on  the  average,  at  20.3  vs.;  and  that — again,  on  the  average — it  does  not 
fall  below  19  or  rise  above  21.6  vs.  The  RL  is  thus  seen  to  be  a  variable, 
not  a  constant  value.  On  the  average,  it  lies  at  20.3  vs.;  but  in  the  indi- 
vidual case  it  is  liable  to  shift  and  displacement,  up  or  down,  by  all  sorts  of 
accidental  influences  affecting  (9's  judgment.  The  average  range  of  these 
influences  is  _j_  1.3. 

Introspections. — Regarded  simply  as  an  exercise  in  psycho- 
physics,  our  experiment  is  now  complete :  we  have  the  RL  and 
its  MV.  It  was,  however,  one  of  the  objects  of  the  experiment 
to  secure  an  exact  introspective  description  of  the  sensations  set 
up  at  and  about  the  point  of  the  qualitative  RL  for  tones.  So  far^ 
the  introspection  required  of  O  has  been  of  the  most  meagre 
kind.  He  has  merely  said  *  Tone,'  '  Doubtful,'  or  *  No  tone  ; ' 
and  his  reports  have  been  symbolised  in  the  record  by  +,  ?, 
and  — .     How  shall  we  obtain  more  detailed  introspections  1 

Three  courses  are  open  to  us.  E  and  O  must  decide  for 
themselves  which  of  these  courses,  or  what  combination  of  them, 
they  prefer,  and  must  give  their  reasons  to  the  Instructor. 

(i)  We  may  so  remodel  the  experiment  as  to  require  an  exact  introspec- 
tive report  from  O  at  every  step  of  every  series.  We  may  instruct  him  to 
say,  first  of  all,  '  Tone  '  or  '  Doubtful '  or  '  No  tone,'  and  then  to  go  on  with 
an  introspective  account  of  what  he  has  heard.  Is  this  course  advisable.? 
(2)  We  may  keep  the  series  as  they  are,  and  require  (9,  in  the  3  min.  pause 
between  series  and  series,  to  sum  up  the  introspective  results  of  the  preced- 
ing experiments.  Is  this  course  adequate  ?  Is  it  advisable  ?  (3)  We  may 
work  through  the  20  series,  and  then,  at  the  conclusion  of  the  quantitative 
work,  take  one  or  more  qualitative  series,  /.  ^.,  series  with  detailed  intro- 
spections at  every  step.  Is  this  course  adequate?  Is  the  extra  work  worth 
while  ? 

Questions. — ( i )  In  what  sense  can  an  average  value  be  said 
to  *  represent'  the  results  of  repeated  measurements  }  What  ad- 
vantage is  there  in  writing  20.3  ±  1.3  for  the  separate  values  on 
p.  8  }     What  disadvantage  is  there  } 

(2)  Could  we  use  any  other  representative  values,  besides  the 
average  and  the  MVf  How  is  the  experimenter  guided  in  hia 
choice  of  representative  values  } 

(3)  Give,  in  your  own  words,  a  psychological  criticism  or  jus- 
tification of  the  method.     We  have  said  *  It  is  best  to  do  this  ' ; 


10  Preliminary  Expet  intents 

*  It  is  inadvisable  to  do  that.'  Show  why  the  various  courses 
proposed  are,  psychologically,  advisable  or  inadvisable. 

(4)  Indicate  the  precise  point,  in  the  method,  at  which  intro- 
spection ends  and  calculation  begins.  What  reason  have  we  for 
changing  from  the  one  to  the  other  ? 

(5)  Do  the  series  furnish  any  other  values,  besides  those  which 
we  have  selected,  for  the  determination  of  the  RL  f 

(6)  What  sources  of  error,  objective  and  subjective,  have  you 
noticed  during  your  performance  of  the  experiment  ?  Do  you 
think  that  the  method  would  be  adequate  to  the  determination 
of  all  sorts  of  RL  f 

(7)  Make  out  a  list  of  the  RL  (or  absolute  limens  in  general) 
which  you  think  a  quantitative  psychology  should  determine.  If 
you  are  interested  in  any  particular  determination,  work  out  the 
details  of  experimentation  (instrument,  method,  etc.),  and  show 
your  plan  to  the  Instructor. 

(8)  Can  you  suggest  other  methods  for  the  determination  of 
the  RL  ? 

EXPERIMENT  H 

§  9.  The  Qualitative  RL  for  Tones:  the  Lowest  Audible  Tone. 

Alternative  Experiment. — The  objects  of  this  experiment  are 
those  of  Exp.  I.,  p.  I. 

Materials,  Adjustment  of  Apparatus,  Preliminary  Ex- 
periments :  see  pp.  i  ff. 

Method. — E  knows,  in  a  general  way,  the  position  of  O's  RL. 
He  is  now  to  determine  it  exactly.  To  this  end,  he  makes  out  a 
series  of  i?-values  such  that  the  highest  value  will  always  call 
forth  the  judgment  *  Tone,'  and  the  lowest  the  judgment  *  No 
tone.'  The  length  and  position  of  this  series  must  be  deter- 
mined in  the  light  of  the  preliminary  experiments.  It  may  cover 
all  possible  values,  from  4  to  24  vs.  ;  it  may  range  between  10 
and  24,  or  between  8  and  20  vs. :  everything  depends  upon  the 
position  of  <9's  RL. 

Having  selected  the  series  of  R  to  be  employed,  ^  ( i )  writes 
the  values  of  these  R  upon  cardboard  tickets,  and  puts  the  tick- 
ets in  a  small  box.  He  (2)  makes  out  a  record-form,  by  writing 
the  /^-values  in  order  along  a  horizontal  line  at  the  head  of  a 


§    lo.      The  Qualitative   TR  for  Tone  II 

sheet  of  cross-ruled  paper.  The  7?  of  a  given  series  are  to  be 
employed  in  haphazard  order,  and  6>'s  judgment  is  to  be  en- 
tered in  its  appropriate  square  upon  the  record-form. 

Experiment. — O  is  seated  in  position  :  his  eyes  closed,  and 
his  unused  ear  stopped  or  open,  as  the  case  may  be.  E  has  his 
record-form  and  the  box  of  tickets  before  him.  The  box  is 
shaken,  to  mix  the  numbers.  E  draws  a  ticket  at  random,  and 
sets  the  lamella  to  the  value  indicated.  Then,  after  the  ready- 
signal  to  (9,  he  says  *  Now ;'  waits  a  short  2  sec. ;  and  actuates 
the  lamella.  (9's  judgment  is  entered,  as  -{-)  —  or  .?  ,  under 
the  corresponding  i?-value  of  the  record-blank.  E  draws  another 
ticket ;  sets  the  lamella  ;  says  *  Now ; '  waits  as  before  ;  takes  a 
second  experiment ;  and  enters  6>'s  judgment  in  the  record.  The 
series  is  continued  until  all  tickets  have  been  taken  from  the  box. 

A  pause,  perhaps  of  3  min.,  is  now  made,  during  which  the 
tickets  are  replaced  in  the  box  and  the  box  thoroughly  shaken. 
Then  a  second  *  Ready !  '  is  given,  and  a  new  series  begins. 

When  5  series  have  been  taken,  E  and  O  change  places.  After 
another  5  series,  they  change  places  again.  Ten  series  are  to  be 
completed  by  each. 

Results. — E  has  his  Table  of  10  series,  which  shows  the  R- 
values  employed  and  6>'s  introspections.  The  Table  should  be 
pasted  or  copied  into  the  note-book. 

The  RL  stands  upon  the  dividing  line  between  noticeableness 
and  unnoticeableness  of  tone.  Hence  we  may  determine  it,  from 
any  series,  by  taking  the  average  of  the  last  unnoticeable  and 
the  first  noticeable  (the  strongest  imperceptible  and  the  weakest 
perceptible)  i? -values.  But  we  have  secured  10  series;  and  we 
have  been  at  this  trouble  because  there  is  safety  in  numbers  :  the 
result  of  a  single  series  might  be  affected  by  all  manner  of  acci- 
dents,— an  uncomfortable  position  on  6^'s  part,  some  outside 
noise,  some  temporary  lapse  of  attention,  etc.  We  proceed, 
accordingly,  to  determine  the  average  value  of  all  20  terms, — of 
the  10  Mast  unnoticeables '  and  the  10  'first  noticeables.'  This 
average  value  fairly  represents  the  RL  for  the  particular  (9,  under 
the  particular  conditions  of  experimentation,  and  of  health,  prac- 
tice, attention,  etc.  We  further  determine  the  MV  ox  AD  of 
the  final  average.     See  p.  8. 


12 


Preliminary  Experiments 


If  the  judgment  corresponding  to  the '  first  noticeable  '  is  a  +,  the  calcu^ 
lation  presents  no  difficulty.  If  it  is  a  ?,  the  question  arises  whether  we  shall 
accept  the  7?-value  for  this  ?,  or  shall  go  higher  in  the  series  and  take  the 
first  + -judgment  as  representing  the  'first  noticeable.'  Everything  de- 
pends upon  the  general  run  of  judgments  within  the  series.  E  should  try 
to  decide  the  matter  for  himself,  and  give  reasons  for  his  decision  to  the 
Instructor. 

Introspections. — See  p.  9. 

EXPERIMENT  III 

§  10.    The  Qualitative  TE  for  Tones:  the  Highest  Audible  Tone. 

— The  object  of  this  experiment  is,  on  the  quantitative  side,  pre- 
cisely to  determine  the  upper  limit  of  tonal  hearing,  and,  on  the 
qualitative,  to  gain  familiarity  with  the  highest  audible  tones. 

Materials. — Edelmann's  Galton  v^histle.  Heavy  standard, 
with  arm  and  clamp.  Small  square  of  felt  or  baize.  [Galton' s 
whistle  is  a  very  small  stopped  labial  pipe.  It  is  actuated  by  the 
squeeze  of  a  rubber  bulb,  and  closed  by  a  piston,  adjustable  by  a 
micrometer  screw.  As  the  piston  is  turned  inwards,  the  pipe  is 
shortened,  and  the  tone  of  the  whistle  rises  in  pitch.  The  num- 
ber of  vibrations  for  any  set  of  the  piston  can  be  calculated  from 
the  reading  of  the  screw.  The  dimensions  of  the  pipe  are  so 
chosen  that  we  can  pass  from  a  clear  shrill  tone 
to  a  mere  noisy  hiss  ;  and  the  problem  is  to 
determine  the  exact  point  at  which  the  R  loses 
its  tonal  component,  and  becomes  noise.  The 
original  form  of  the  Galton  whistle  is  shown  in 
Fig.  3- 

The  Edelmann  whistle  (Fig.  4)  is  composed  of 
two  separate  and  separately  adjustable  parts  : 
the  body  of  the  pipe,  with  its  piston,  and  the 
mouth-piece.  The  pipe  is  a  cyHndrical  closed 
pipe,  whose  open  end  lies  with  its  sharp  circular 
edge  directly  over  against  the  mouth-piece. 
The  cylindrical  lip  suggests  the  principle  of  the 
steam  whistle  rather  than  that  of  the  organ  pipe. 
The  length  of  the  pipe  can  be  read  from  the 
scale  of  the  instrument  to  tenths,  and  by  the 
-eye  to  hundredths,  of  i  mm.  The  mouth-piece  is  also  circular: 
the  air  is  forced,  by  pressure  on  the  rubber  bulb,  through  an  an- 


FlG.  3. 


§    10.      The  Qualitative   TR  for  Tone 


13 


nular  opening  or  wind-way ;  impinges  on  the  lip  of  the  pipe  ;  and 
throws  the  contained  air-column  into  strong  vibration.  The  width 
of  the  mouth  of  the  pipe  (the  distance  separating  lip  and  mouth- 
piece) can  be  regulated  by  a  second  micrometer  screw,  reading 
accurately  to  tenths  of  i  mm.  The  Edelmann  whistle,  therefore, 
allows  the  tone  to  be  treated  as  a  function,  not  only  of  the  length 
of  the  pipe,  but  of  this  and  of  the  width  of  the  mouth  as  well.] 
Adjustment  of  Apparatus. — E  wraps  the  felt  about  the 
heavy  plate  of  metal  that  connects  the  two  halves  of  the  whistle, 
and  clamps  it  firmly  in  the  arm  of  the  standard.  The  whistle 
lies  horizontally  in  the  clamp,  the  body  of  the  pipe  ta  ^'s  left. 
O  sits  as  near  the  instrument  as  is  convenient  (the  distance  is  to 

be  kept  constant  through- 
out the  experiment),  his 
better  ear  turned  towards 
it.  A  chin-rest  may  be 
used,  to  ensure  constancy 
of  position.  The  unused 
ear  may  be  stopped  or 
open,  according  to  circum- 
stances :  see  p.  2  above. 

Preliminary  Experi- 
ments.— O  must  have  pre- 
liminary practice  in  the 
hearing  of  highest  tones, 
in  order  that  he  may  ac- 
quire a  standard  of  judg- 
ment. E  must  also  have 
preliminary  practice  in  the 
manipulation  of  the  whistle. 
It  will  be  advisable  to 
begin  6^'s  practice  with  an 
R  that  gives  a  clear  tone. 
By  working  upwards,  from 
this  lower  limit,  E  will 
be  able  to  establish,  in  a 
rough  and  tentative  way,  the  position  of  (9's  TR.  E  sets  the 
whistle  by  help  of  a  Table  furnished  by  the  Instructor :  the  unit 


14  Preliminary  Experiments 

of  change  is  looo  vs.  O  knows  this  ;  but  is  to  know  nothing  of 
the  result,  and  as  little  as  possible  of  the  course  of  the  prelim- 
inary experiments.  His  eyes  are  closed  throughout.  He  knows 
that  E  is  working  from  below  upwards  ;  but  that  is  all.  E  may 
advance  lOOO  vs.  at  each  step,  or  may  sound  the  same  R  half-a- 
dozen  times  over, — going  through  the  motions  of  readjustment, 
but  not  actually  readjusting  the  whistle. 

E  will  probably  find  that  the  experiment  can  be  performed 
with  a  single  width  of  the  mouth  of  the  whistle.  The  manipula- 
tions are  then  reduced  to  the  setting  of  the  piston  with  the  left 
hand,  and  the  squeezing  of  the  rubber  bulb  with  the  right.  The 
stimulus  should  be  given  by  a  squeeze  only,  not  by  squeeze  and 
release  of  the  bulb.  The  bulb  is  held  in  the  hollow  of  the  hand, 
and  pressed  sharply  with  the  thumb. 

Method. — See  pp.  3  ff.,  above.  The  first  of  the  20  series  is, 
in  this  case,  an  |  series. 

Experiment,  Results,  Introspections.  —  See  pp.  6  ff., 
above. 

Questions. — (i)  What  sources  of  error,  objective  and  sub- 
jective, have  you  noticed  during  your  performance  of  the  experi- 
ment 1  Do  you  think  that  the  method  would  be  adequate  to  the 
determination  of  all  sorts  of  TR  f 

(2)  Make  out  a  list  of  the  TR  that  you  think  a  quantitative' 
psychology  should  determine.     If  you  are  interested  in  any  par- 
ticular determination,  work  out  the  details  of  experimentation 
(instrument,  method,  etc.),  and  show  your  plan  to  the  Instructor. 

(3)  Write  out  an  introspective  comparison  of  lowest  and  high- 
est tones.  Write  out  a  similar  comparison  of  the  noises  given 
by  the  lamella  and  the  whistle,  below  and  above  the  limits  of 
tonal  hearing. 

§  II.  The  Intensive  RL  for  Pressure. — The  object  of  this  ex- 
periment is  to  determine  the  least  pressure  that  can  be  perceived, 
under  given  conditions,  by  the  resting  skin. 

EXPERIMENT  IV 

(i)  Materials. — Scripture's  touch-weights.  [A  full  set  of 
touch-weights  consists  of  20  discs  of  elder-pith  or  cork,  3mm.  in 


§    II.      The  Intensive  RL  for  Pressure 


15 


diam.,  suspended  by  silk  threads  from  short  wooden  handles. 
The  weights  range  between  i  and  10  mg.  with  differences  of  i 
mg.,  and  between  10  and  30  mg.  with  differences  of  2  mg.] 


f=^ 


^ 


Fig.  5. 


Manipulation. — Convenient  parts  of  the  skin,  for  this  experi- 
ment, are  temple,  middle  of  forehead,  side  of  nose,  inner  surface 
of  wrist  and  forearm,  finger-tip.  O  must  sit  or  lie  comfortably, 
in  a  constant  position ;  the  temperature  of  the  room  should  be 
kept  constant. 

E  applies  the  weights  normally  to  the  surface  to  be  experi- 
mented on.  They  must  be  set  down  steadily  and  evenly  ;  left 
for  2  sec. ;  and  then  carefully  removed.  A  slight  impact  will 
lower  the  RL  beyond  its  proper  value.  If  a  weight  be  set  down 
unevenly,  or  the  thread  twitched  while  the  weight  lies  upon  the 
skin,  a  tickling  will  be  produced.  A  jerk  away,  at  the  end  of  the 
2  sec,  will  have  the  same  effect  as  an  initial  impact.  Some  little 
practice  is  required  before  the  weights  can  be  applied  and  re- 
moved in  a  satisfactory  manner. 

Method. — The  place  of  the  RL  is  roughly  determined  in  pre- 
liminary experiments.  The  method  is  that  of  Exp.  II.,  pp.  10 
ff.,  above. 

EXPERIMENT  V  * 

(2)  Materials. — von  Frey's  hair  aesthesiometer.  [The  in- 
strument consists  essentially  of  a  horsehair  or  human  hair,  attached 
to  a  fine  wire,  which  slides  in  a  metal  tube  of  very  small  bore. 
The  projecting  length  of  hair  can  be  varied  at  will,  and  the  tube 


i6 


Preliminary  Experiments 


clamped  at  any  required  point  by  a  screw.  A  mm.  scale,  en- 
graved on  the  tube,  makes  it  possible  to  set  the  instrument  at  the 
same  point  in  different  experiments.] 


< 


rTYJ'\ 


D 


Fig.  6. 

Preliminary  Experiments. — Find  a  responsive  pressure 
spot,  on  the  back  of  the  hand,  and  mark  its  position  by  a  ring  of 
dye.  Assure  yourself  that  the  spot  does  not  reply  to  stimulation 
by  the  longest  length  of  hair  afforded  by  the  aesthesiometer,  and 
that  it  replies  clearly  to  the  shortest  length.  Determine  roughly, 
in  a  I  series,  the  position  of  6^'s  RL. 

Suppose  that  a  hair  is  waxed  at  right  angles  to  the  end  of  a  light 
wooden  handle,  and  that  the  point  of  the  hair  is  set  down  perpendicularly 
upon  some  point  of  the  skin  (Fig.  7,  A).     As  the  handle  is  lowered,  and  the 


Fig.  7. 

Application  of  a  hair  to  the  cutaneous   surface.     S,  skin ;  /*,  point  of  inflection 

of  the  S-shaped  curve. 

pressure  consequently  increased,  the  hair  bends  to  form  an  S-shaped  curve 
(Fig.  7,  B).  So  long  as  the  point  of  inflection  of  the  curve  lies  on  the  per- 
pendicular whjch  passes  through  the  point  of  stimulation,  the  hair  is  pressing 
vertically  upon  the  skin.  As  soon  as  the  hair  gives  to  one  side,  and  the 
point  of  inflection  leaves  the  perpendicular,  a  lateral  push  is  added  to 
the  vertical  pressure. 

If,  now,  the  hair  is  set  down  upon  the  pan  of  a  sensitive  balance,  instead 
of  upon  the  skin,  it  is  found  to  exert  its  maximal  pressure  (to  compensate 
the  heaviest  weight)  just  before  it  gives.     The  hair  saves  itself,  so  to  speak, 


§    II.      The  Intensive  RL  for  Pressure 


17 


by  yielding  and  turning,  before  the  limit  of  its  elasticity  is  transcended. 
Moreover,  its  maximal  pressure  remains  constant  over  a  long  period  of  time, 
and  in  spite  of  much  work.  These  properties  render  it  an  admirable  in- 
strument for  cutaneous  stimulation. 

In  applying  the  hair  to  the  skin,  E  must,  however,  be  careful  not  to  push 
the  curvature  too  far.  If  the  curve  of  flexure  is  a  pronounced  6",  the  pres- 
sure of  the  hair  will,  it  is  true,  be  still  vertical,  but  it  will  not  be  evenly 
distributed  over  the  whole  surface  of  the  cross-section :  the  edge  of  the  hair 
will  dig  into  the  skin.  Fortunately,  the  pressure  remains  constant,  at  a 
nearly  maximal  value,  with  a  very  slight  curving  of  the  hair ;  and  it  is, 
accordingly,  this  slight  bend  which  should  be  employed  in  practice.   . 

The  series  may  be  taken  in  units  of  the  mm.  scale,  and  the  pressure 
values  obtained  from  the  Instructor  after  the  completion  of  the  experiment. 

Care  must  be  taken  to  apply  the  hair  always  to  the  same  spot,  and  (so  far 
as  possible)  with  the  same  rate  of  movement.  Long  hairs  quiver  and  are 
apt  to  slip,  and  stout  hairs  are  apt  to  spring  as  they  are  lifted  from  the  skin, 
so  that  careless  manipulation  will  lead  to  tickling  or  to  the  stimulation  of 
neighbouring  pressure  spots.  The  hair  need  not  be  held  on  the  skin  for  more 
than  I  sec.  The  pressure  spots  are  very  readily  fatigued  ;  intervals  of  at 
least  30  sec.  must  elapse  between  stimulations,  and  E  and  O  should  change 
places  after  every  one  or  two  series. 

Method. — The  strict  serial  method  of  Exp.  I.  or  the  haphaz- 
ard arrangement  ©f  Exp.  II.  may  be  followed. 


EXPERIMENT  VI 

(3)  Materials. — von  Frey's  limen  gauge,  with  discs,  springs, 


Fig.  8. 

and  kymograph  attachments.     [The  limen  gauge,  Fig.  8,  consists 
essentially  of  two  levers,  L  and  L',  which  turn  about"  parallel 

B 


i8 


Preliminary  Experiments 


axes,  A  and  A\  and  are  coupled  by  a  clock-spring.  The  lower 
lever,  Z,  carries  at  its  extremity  a  point  of  horn  or  bone.  This 
is  the  stimulus  point  :  it  rests  upon  a  small  disc  of  cork  or  card- 
board, which  in  turn  rests  upon  the  skin.  Attached  to  the  upper 
axle,  ^',  is  a  paper  scale,  reading  in  degrees  from  o°  to  50°. 
The  horizontal  arm  is  clamped  in  a  standard  ;  the  clamp  serves 
for  the  coarse  adjustment  of  the  instrument,  while  the  fine  ad- 
justment is  effected  by  the  nut  N.  The  intensity  of  stimulation 
is  regulated  by  the  set-screw  5  and  read  off  (in  degrees)  from 
the  paper  scale.     Six  clock-springs  of  different  tension,  and  discs 


Fig.  9,  a. 


of  three  sizes  (0.75,  i,  1.5  mm.  diam.),  are  supplied  with  the 
gauge. 

As  thus  described,  the  apparatus  permits  us  to  vary  the  place, 
the  intensity  and  the  area  of  the  stimulus.  The  RL  depends, 
further,  upon  its  rate  of  application.  To  control  this  factor,  we 
have  recourse  to  the  following  device  (Fig.  9,  a^  b). 

A  thread  attached  to  the  end  of  the  lever  Z'  is  led  upwards, 
over  a  small  and  easily  running  pulley  of  hard  wood,  P,  to  the 
inner  end  of  the  metal  strip  M.     The  strip  is  supported  hori- 


§    II.      The  Intensive  RL  for  Pressure 


19 


zon tally  by  the  adjustable  bar  B^  and  turns  freely  about  its  ver- 
tical axis  at  C.  The  outer,  free  end  of  M  lies  directly  over  the 
upper  edge  of  a  kymograph  cylinder,  which  carries  the  rounded 
pin  G.  It  is  clear  that,  if  the  kymograph  is  started,  the  pin  G 
will,  once  in  every  revolution,  strike  the  free  end  of  M,  take  it  a 
certain  distance,  and  presently  release  it.     So  long  as  M  is  mov- 


FiG.  9,  b. 

ing,  an  upward  pull  is  exerted  upon  the  lever  L\  and  the  stimu- 
lus point  of  L  is  pressed  down  upon  its  disc.  As  soon  as  M  is 
released,  the  apparatus  must  come  back  with  a  jerk  to  its  position 
of  equilibrium. 

The  jerk  would,  however,  be  a  source  of  serious  disturbance  in 
liminal  determinations.  To  avoid  it,  we  run  along  the  side  of  B^ 
beneath  M,  a  metal  rod  upon  which  slides  the  follower  F.  A 
thread  passes  from  F^  over  a  pulley,  to  the  little  weight  W.  As  M 
moves,  F  (drawn  forward  by  the  weight)  gently  follows  it  up ;  so 
that,  when  M  is  released  by  the  pin  (9,  the  apparatus  remains  in 
its  new  position,  so  to  say,  as  a  matter  of  course,  without  the 
least  jar  or  recoil.  To  reset  it,  E  has  merely  to  move  F  back 
with  his  finger  to  its  original  starting-point. 

The  bar  B  is  adjusted  vertically  by  means  of  the  clamp  which 
holds  it  to  the  standard.  It  consists  of  two  squared  rods,  fitted 
with  spline  and  feather,  and  connected  by  the  broad  clamp  D. 


20  Preliminary  Experiments 

This  complication  is  introduced  in  order  that  we  may,  without 
manipulating  the  limen  gauge  itself,  vary  the  intensity  as  well  as 
the  rate  of  application  of  the  pressure  stimulus.  The  farther  M 
projects  over  the  kymograph  cylinder,  the  farther,  of  course,  will 
G  take  it,  and  the  more  severe,  accordingly,  will  be  the  final 
pressure  of  Z.  First  of  all,  then,  in  arranging  an  experiment, 
we  clamp  the  upper  rod  of  B  (which  carries  a  zero-point)  at  a 
convenient  distance  from  the  kymograph  :  this  is  the  coarse 
adjustment.  Afterwards,  by  loosening  D^  and  moving  the  lower 
rod  of  B  (which  carries  a  scale)  back  or  forth  along  the  upper 
rod,  we  are  enabled  to  take  small-stepped  pressure  series,  \  or 
|,  as  the  experiment  requires  :  this  is  the  fine  adjustment.  By 
changing  the  rate  of  revolution  of  the  drum,  we  change  the  rate 
of  application  of  the  stimulus ;  by  shifting  M  in  or  out,  we  vary 
its  Hmiting  intensity.] 

Preliminaries. — (i)  We  must  know  the  rate  of  rotation  of 
the  cylinder,  at  the  various  speeds  of  which  the  kymograph  is 
capable.  The  drum  is  covered  with  smoked  paper,  and  half-a- 
dozen  records  are  obtained,  at  short  intervals,  from  a  suitable 
time  marker. 

Before  the  paper  is  varnished,  mark  upon  it  {a)  the  setting  of  the  kymo- 
graph, ip)  the  date,  {c)  the  times  of  the  separate  records,  and  {d)  the  average 
time  of  revolution  with  its  MV.  If  the  instrument  is  well  made,  the  parts 
kept  oiled  and  free  from  dust,  and  the  clockwork  wound  up  regularly,  this 
J/Fwill  be  negligibly  small.  The  test  should,  however,  be  repeated  at  fre- 
quent intervals.  The  dated  records  are  to  be  kept  in  a  special  drawer  in 
the  laboratory,  where  they  will  be  available  for  comparison. 

(2)  Preliminary  experiments  must  be  made  with  various  com- 
binations of  clock-springs  and  kymograph  rates,  to  determine  the 
approximate  position  of  the  RL.  The  experiments  should  be 
conducted  with  some  care,  since  the  skin  easily  becomes  fatigued, 
and  it  is  therefore  essential  that  the  regular  experimental  series 
be  kept  short. 

Good  places  to  work  upon  are  the  ball  of  the  thumb  and  the  volar  surface 
of  the  wrist.  O  must  above  all  things  be  comfortable  :  if  his  arm  begins  to 
tingle,  or  if  he  feels  his  position  to  be  in  the  least  degree  constrained,  the 
experiment  must  be  broken  off.  His  hand  and  forearm  should  rest  in  a 
plaster  mould,  or  (what  does  as  well)  in  a  shallow  box  of  sand,  covered  with 
a  light  cloth,  and  packed  to  the  right  shape. 


§11.      The  Intensive  RL  for  Pressure  2 1 

(3)  In  the  actual  experiments,  all  adjustments  are  made  by 
means  of  the  bar  B^  but  all  readings  are  taken  (in  degrees)  from 
the  scale  of  the  limen  gauge.  It  is  necessary,  therefore,  to  work 
out  a  correlation  of  the  two  scales.  To  this  end,  the  lever  L  is 
set  at  o,  while  the  stimulus  point  rests  upon  a  plate  of  glass  or 
metal ;  and  B  is  adjusted,  in  successive  trials,  until  the  required 
scale  of  pressure  values  (degree  readings)  has  been  obtained. 

If  the  adjustments  of  B  are  accurately  made,  and  if  the  instrument  is 
kept  in  good  working  order,  the  two  scales  will  remain  in  perfect  correlation ; 
so  that  E  may  confidently  assume  that  a  given  setting  of  B  will  reproduce  a 
required  scale-reading  for  L.  If  by  any  accident  the  adjustment  of  B  is  in- 
accurate, the  effect  is  simply  to  produce  a  slight  irregularity  in  the  size  of 
the  steps  within  a  series.  Since  E  takes  his  readings  from  the  lower  scale, 
the  irregularity  will  be  duly  entered  in  the  record  of  the  experiment. 

Method. — The  strict  serial  method  of  Exp.  I.,  or  the  hap- 
hazard method  of  Exp.  II.  may  be  followed. 

Results. — The  scale  readings  of  the  limen  gauge  must  be 
translated  into  terms  of  pressure.  The  translation  is  done, 
very  simply,  by  substituting  for  6^'s  hand  the  pan  of  a  balance, 
and  finding  the  weight  that  will  just  compensate  the  pressure  of 
the  stimulus  point.  When  the  critical  values  of  the  series  have 
thus  been  determined  in  gr.,  E  calculates  the  average  and  its  MV 
in  the  ordinary  way. 

The  apparatus  which  we  have  here  employed  allows  us  to  vary  the  applica- 
tion of  the  R  in  four  different  ways.  We  may  vary  {a)  the  intensity  of  the  R  : 
this  is  necessary  to  the  liminal  determination,  and  does  not  come  further  into 
consideration.  We  may  vary  {b)  the  place  of  application,  and  so  work  out  a 
comparative  series  of  limens  for  different  parts  of  the  skin.  We  may  vary 
{c)  the  extent  of  the  R,  and  so  work  out  a  comparative  series  of  limens  in 
terms  of  the  area  stimulated.  Finally,  we  may  vary  {d)  the  rate  of  applica- 
tion, and  so  work  out  a  comparative  series  of  limens  in  terms  of  gr.  to  the 
I  sec,  or  gr./sec. 

As  regards  {b),  it  is  only  necessary  to  say  that  we  have  chosen  hairless 
parts  of  the  skin  (palm  of  hand,  volar  surface  of  wrist)  in  order  to  avoid 
the  complications  that  arise,  even  after  careful  shaving,  from  the  presence 
of  hairs  beneath  the  disc.  As  regards  (^),  comparative  determinations  are 
difficult,  owing  to  the  irregular  distribution  of  the  pressure  spots.  E  must 
find  an  area  of  O's  skin  where  there  is  a  sensitive  spot  surrounded  by  spots 
of  markedly  lower  sensitivity.  The  centre  of  the  disc  employed  must 
always  cover   the  sensitive  spot.     The   stimulus   point   is  applied   to  the 


22  Preliminary  Experiments 

centre  of   the    disc ;  and  the  disc  itself  must  He  evenly  and  smoothly  upon 
the  skin. 

As  regards  {d)  we  must  go  into  a  little  more  detail.  We  are  to  work  out 
a  comparative  series  of  limens  in  terms  of  gr.  to  the  i  sec.  We  have,  then, 
to  take  account  both  of  the  tension  of  the  clock-spring  and  of  the  rate  of 
revolution  of  the  drum.  Suppose,  first,  that  the  drum,  revolving  at  its 
slowest  rate,  makes  one  revolution  in  85.7  sec,  /.  ^.,  that  it  travels  4.2°  in 
the  I  sec;  and  that,  revolving  at  its  fastest  rate,  it  makes  one  revolution 
in  9.6  sec,  /.  ^.,  travels  37.5°  in  the  i  sec  Here  is  the  one  set  of  data. 
Now  let  us  bring  the  stimulus  point,  with  the  lever  L  at  o,  over  the  pan  of  a 
balance,  and  lower  the  limen  gauge  until  the  point  just  touches  the  pan :  let 
us  place  a  i  gr.  weight  in  the  other  pan  of  the  balance :  and  let  us  screw 
down  the  set-screw  S^  until  the  pointer  of  the  balance  marks  o.  We  can 
read  off  from  the  scale,  in  degrees,  the  tension  of  the  spring  for  i  gr.  Sup- 
pose that,  with  the  weakest  of  the  six  clock-springs,  the  scale  reading  is  30°, 
and  with  the  strongest  2.3°.  Here  is  the  other  set  of  data.  Putting  the 
two  together  we   infer  that  with  the  slowest  revolution  and  the  weakest 

spring  the  stimulus  point  is  applied  to  the  disc  at  the  rate  of  —  or  0.14 

gr./sec;  while  with  the  quickest  revolution  and  the  strongest  spring  it  is  ap- 

plied  at  the  rate  of  ^-!-^  or  16.3  gr./sec    Other  determinations  are,  of  course, 

2.3 
to  be  made  for  the  remaining  combinations.     Variation  in  the  rate  of  ap- 
plication of  the  R  will    then  be  expressed  in  the  results  by  a  column  of 
figures  under  the  rubric  gr./sec,  which  vary  (in  the  supposed  case)  between 
the  extreme  limits  0.14  and  16.3. 

Questions. — (i)  What  sources  of  error,  subjective  and 
objective,  have  you  noticed  during  your  performance  of  the 
experiment  ?     How  have  you  attempted  to  avoid  them  ? 

(2)  Compare  any  one  of  the  three  experiments  of  this  Section 
with  Exp.  I.,  II.  or  III.  Which  is  the  more  difhcult .?  Answer 
the  Question  both  on  the  objective  and  on  the  subjective  side : 
i,  e.y  in  terms  both  of  manipulation  of  instruments  and  of 
sensory  judgment.     Give  full  reasons  for  your  answer. 

(3)  Suggest  method  and  materials  for  the  determination  of 
the  intensive  RL  of  pain,  warmth,  and  cold.  Try  to  think  your- 
self into  the  experiments,  and  meet  the  imagined  difficulties  of 
the  procedure  one  by  one,  as  they  occur  to  you. 

§  12.  The  Intensive  RL  for  Sound.— The  object  of  this  ex, 
periment  is  to  determine  the  least  sound  that  can  be  perceived 
by  the  single  ear. 


§  12.      The  Intensive  RL  for  Sound  23 

EXPERIMENT  VII 

(i)  Materials. — Watch,  in  padded  box,  with  (separate) 
padded  front.  Wooden  uprights  and  tape  (15  m.).  Metre  rod. 
Ear  plugs.     Eye-shade. 

Adjustment  of  Apparatus. — The  experiment  is  to  be  per- 
formed in  a  corridor  or  long  room,  which  is  free  from  noise, 
and  whose  furniture  is  always  arranged  in  the  same  way.  O  sits 
sidewise  to  the  length  of  the  room  :  his  eyes  are  shaded,  and  his 
unused  ear  closed.  The  uprights  are  placed  along  the  room,  in 
a  line  with  the  line  that  joins  (9's  ears  ;  the  tape  is  stretched 
across  the  tops  of  the  uprights,  and  marked  off  into  0.5  m.  divisions. 
The  watch,  lying  in  the  box  with  its  face  turned  directly  toward 
<7's  ear,  is  carried  in  and  out  above  the  tape  by  E ;  it  should  be 
held  constantly  at  the  level  of  the  ear,  and  should  be  moved  in  a 
line  a  little  forward  from  the  line  passing  through  6>'s  two  ears. 
It  is  convenient  to  stretch  the  tape  at  such  a  height  from 
the  floor  that  ^'s  elbow  just  grazes  it  when  the  watch  is  held 
in  the  proper  position. 

Preliminary  Experiments. — Suppose  that  (9's  left  ear  is  to 
be  tested.  E  takes  the  box  in  his  left,  and  the  box  cover  in  his 
right  hand.  Starting  from  the  first  metre  mark,  he  moves  out- 
wards, by  metre  intervals.  At  every  stop,  the  usual  *  Now  ! '  is 
given,  and  the  box  uncovered  for  5  sec.  O  signals,  by  a  move- 
ment of  his  hand,  whether  or  not  he  hears  the  ticking.  Blank 
experiments  should  be  introduced,  here  and  there ;  all  care  must 
be  taken  that  O  shall  not  know  of  their  occurrence. 

The  test  is  repeated,  in  the  opposite  direction.  The  experi- 
ments, outgoing  and  incoming,  are  to  be  continued  until  the 
position  of  (9's  RL  has  been  roughly  determined. 

Method. — The  strict  serial  method  of  Exp.  I.  or  the  hap- 
hazard arrangement  of  Exp.  II.  may  be  followed.  Unit  of 
change,  0.5  m. 

EXPERIMENT   VIII 

(2)  Materials. — Lehmann's  acoumeter.  [The  acoumeter 
consists  of  a  wooden  platform,  supported  by  three  set-screws  5  (in 
Fig.  10  only  the  lower  half  of  one  of  the  two  right-hand  screws 
is  shown),  at  the  centre  of  which  a  stout  mm.  screw  M  is  set 
vertically  in  a  screw-nut.     The  head  of  the  mm.  screw  is  divided 


24  Preliminary  Experiments 

into  quarters;  a  mm.  scale  stands  beside  it.  A  small  spring 
forceps  F  lies  upon  the  head  of  the  mm.  screw;  it  is  kept  in 
place  by  a  pin,  which  fits  into  a  corresponding  hole  at  the  centre 


Fig.  io. 

of  the  screw-head.  At  the  front  end  of  the  platform  is  a  padded 
trough ;  the  padding  is  carried  backward  over  a  cross-strip  of 
wood,  W^  whose  oblique  surface  lies  directly  beneath  the  jaws  of 
the  forceps.  A  shot,  dropped  from  the  forceps,  falls  vertically 
upon  a  square  of  glass,  copper  or  cardboard,  laid  upon  the  surface 
of  the  strip,  and  rebounds  into  the  trough.  The  noise  thus  pro- 
duced constitutes  the  stimulus ;  its  intensity  is  expressed  as  the 
product  of  the  weight  of  the  shot  into  its  height  of  fall,  /.  e.y  in 
mg.-mm.] 

Preliminary  Experiments. — O  sits  at  a  fixed  distance  of  lo 
m.  from  the  acoumeter,  his  unused  ear  plugged  and  his  eyes 
shaded.  E  is  to  determine  the  position  of  (9's  RL  for  this  fixed 
distance  and  with  a  shot  of  known  weight.  He  varies  the  height 
of  fall  by  steps  of  i  mm.,  in  \  and  \  series,  until  the  critical 
height  has  been  roughly  ascertained. 

The  zero-point  of  the  vertical  mm.  scale  lies  in  the  same  horizontal  line 
as  the  point  of  impact  upon  the  square  :  all  three  squares  have  sides  of  i 
cm.  and  a  thickness  of  i  mm.  To  take  the  initial  scale-reading,  E  turns  the 
forceps  round  until  its  jaws  strike  the  scale,  and  lays  a  narrow  strip  of  white 
cardboard  against  the  face  of  the  scale  and  the  lower  surface  of  the  forceps. 
No  further  reading  is  necessary,  as  count  can  be  kept  of  the  turns  of  the 
screw-head.     The  same  strip  of  cardboard  is  useful  for  regulating  the  posi- 


§  13-      Weber's  Law  25 

tion  of  the  shot.  E  places  the  shot  in  the  forceps  ;  rests  the  jaws  of  the 
forceps  on  the  card  held  horizontally  ;  releases  the  shot,  so  that  it  settles 
down  upon  the  card  ;  and  lets  the  jaws  of  the  forceps  close  again.  He  ad- 
justs in  this  way  for  every  test.  An  extra  forceps  is  useful,  for  picking  up 
the  fallen  shot. 

Some  little  practice  is  required  in  the  manipulation  of  the  forceps.  The 
pinch  of  thumb  and  finger  must  be  firm  and  uniform ;  the  downward  pres- 
sure must  not  be  so  strong  that  the  forceps  scrapes  against  the  screw-head. 

Method. — The  strict  serial  method  of  Exp.  I.,  or  the  haphaz- 
ard arrangement  of  Exp.  II.,  may  be  followed.  The  unit  of 
change  must  be  determined  from  the  preliminary  experiments ; 
it  is  probable  that  a  step  of  0.5  mm.  will  be  small  enough. 

Questions. — O  (i)  Sometimes,  in  the  course  of  these  experi- 
ments, you  are  quite  sure  that  you  hear  the  sound ;  sometimes, 
you  say  with  confidence  that  you  only  *  imagine  '  it.  How  do  you 
distinguish  between  a  real  and  an  imagined  sound } 

O  {2)  Have  you  had  any  similar  experience  in  preceding  ex- 
periments .?     Is  it  present  to  the  same  degree  in  all  t     Why .? 

E  and  O  (3)  Can  you  suggest  experiments  that  would  make 
it  difficult  for  you  to  decide  whether  or  not  an  objective  stimulus 
had  been  given,  and  that  would  therefore  bring  out  clearly  the 
criteria  which  you  apply  in  distinguishing  between  a  real  and  an 
imagined  impression } 

§  1 3.  Weber's  Law. — We  said  above  (p.  xxxiv)  that  the  formula 
of  correlation  between  i? -magnitude  and  sense-distance  has,  in 
certain  sense  departments,  been  worked  out  with  some  degree  of 
fullness.  We  said,  also,  that  the  formula  has  been  worked  out 
in  terms  of  two  different  kinds  of  sense-distances :  in  terms  of 
supraliminal  distances,  arbitrary  distance-units  that  are  larger 
than  the  DL  ;  and  also  in  terms  of  liminal  distances,  i,  e.,  of  the 
DL  or  the  j.  n.  d.  itself  (p.xxxv).  In  the  former  case,  it  has  been 
worked  out  most  successfully  for  brightnesses ;  in  the  second,  it 
has  been  verified  for  a  large  number  of  different  sensations. 
We  have  now  to  see  what  it  is  and  how  it  is  derived. 

(i)  Suppose  that  we  have  before  us  a  series  of  50  grey  papers, 
delicately  graded  from  dark  to  light.  We  are  required  to  divide 
up  this  practically  continuous  series  into  6  equal  sense-distances. 
Keeping  the  first  and  the  last,  the  darkest  and  the  lightest  papers, 


26  Preliminary  Experiments 

we  are  to  pick  out  5  other  greys  between  them,  at  such  points 
that  the  sense-distance  G^ 6^2=  ^2 <^3=  G^ G\,  etc.  With  a  little 
practice,  the  task  is  by  no  means  difficult. 

Having  chosen  our  7  papers,  6^1,  6^2,  ^3,  G^y  Gr^^  G^e,  G-j^  we 
determine  their  photometric  values,  their  physical  light-values  or 
their  objective  luminosities.      We  find  that  these  photometric 


Fig.  II. 

Graphic  representation  of  the  logarithmic  relation  obtaining  between  intensive 
pressure  sensation  S  and  its  adequate  stimulus  R  (Weber's  Law).  From 
A.  Hofler,  Psychologie,  1897,  138. — For  the  method  of  drawing  the  curve,  see 
§  15  below. 

values  form  (approximately)  a  series  of  equal  quotients.     That 

.  .     are 


equal,  then  the  i?-quotients  %    %    % 

Crl         O2         G3 


is  to  say,   if  the  sense-distances   6^16^2,  G2GS,  GsG^, 

.    are  also  equal. 

An   arithmetical   series   of   sense-distances    is   paralleled    by  a 
geometrical  series  of  corresponding  i?-values. 

That  is  our  first  general  result.  Now  (2)  suppose  that  we 
are  determining  the  DL  for  some  given  sensation  (pressure,  or 
sound,  or  smell) ;  and  that  we  make  our  determination  at  various 
parts  of  the  intensive  scale  (e.g:,  with  25,  50,  100,  200  gr.).  We 
find,  again,  that  the  j.  n.  sense-distances  correspond  to  i?-incre. 
ments  that  are,  approximately,  equal  fractions  of  the  original  R. 

If   we    call   the    i?-increments    A^,    then    the    quotients    =— \ 

A^2      A/?3 

^2  '     ^3  ' 


.  are  approximately  equal.     So  that,  in  this  case 


also,  an  arithmetical  series  of  (least)  sense-distances  is  paralleled 
by  a  geometrical  R-ser'iQS. 

That  is  our  second  general  result.  Putting  the  two  together, 
we  may  say  that  any  progressive  series  of  equal  intensive  sense- 
distances  is  paralleled,  on  the  physical  side,  by  an  approximately 


§13.      IVeder's  Law 


27 


geometrical  series  of  i? -values.  Our  next  problem  is  to  gen- 
eralise this  statement.  Ify  we  have  found,  the  vS-distances  form 
such  and  such  a  series,  then  the  i?-values  form  such  and  such 
another  series.  But  what  is  the  general  correlation,  that  holds 
irrespectively  of  'if  and  ' then ' .? 

(3)  We  can  work  out  our  formula,  most  easily,  in  terms  of 
supraliminal  distances.  We  know  from  (i)  that  the  magnitude 
of  any  sense-distance  is  in  some  way — in  what  way  we  have  to 


Graphic  representation  of  the  correlation  of  tonal  pitch  (Si,  S2,  .  .  .  are  tones 
an  octave  apart)  with  vibration  rate  (R).  If  we  assume  the  sensible  equality  of  the 
octave-distance,  at  various  parts  of  the  tonal  scale,  the  relation  is  similar  to  that 
obtaining  between  the  .S"  and  R  that  fall  under  Weber's  Law.  From  A.  Hofler, 
Psych.,  1897,  135. 

discover — dependent  upon  the  quotient  of  the  two  R  which  limit 
it.  Let  this  unknown  dependency  be  expressed  by  the  mathe- 
matical sign  of  function,/.  Then  we  have,  for  two  successive 
sense-distances,  the  equations  : 

Adding  these  equations,  we  get : 


28  Preliminary  Experiments 

But  we  know  from  ( i )  that : 

and,  of  course, 

We  have,  then,  finally  : 

/(l)+/©=/(l)(^)- 

Now  the  only  continuous  function  that  can  satisfy  an  equation  of 
this  form  is  (as  we  learn  from  the  mathematical  text-books)  a 
logarithmic  function.  Hence  we  may  write  (inserting  a  constant 
factor,  Cy  to  indicate  our  choice  of  some  particular  logarithmic 
system) : 

^1 


S^S,^  c  log  ^S 

3733  =  c  log  ^2. 

Or,  in  general,  if  S^  and  R^  denote  the  5  and  R  with  which  we 
start,  and  S  and  R  themselves  denote  any  other  sensation  and 
its  corresponding  stimulus  value  : 


R 


^  ^o  =  ^  log  ^ 

And,  lastly,  if  we  denote  the  intensive  5-distances  reckoned  from 
an  initial  S^  by  S,  and  the  i?-intensities  calculated  in  terms  of 
the  corresponding  R^  by  R,  we  have  simply  : 

S  =  ^  log  R 

This,  then,  S  =  ^  log  R,  is  the  general  formula  required.  It 
represents  Fechner's  formulation  of  what  is  known  as  Weber's 
Law  (p.  xxxviii  above).  Weber  had  declared  in  1 834,  on  the  basis 
of  experiments  with  weights  and  visual  distances  which  seemed  to 
establish  a  constancy  of  the  relative  DL^  that  in  the  act  of  com- 
parison of  two  stimuli  the  object  of  perception  is  not  the  differ- 
ence between  the  stimuli,  but  rather  the  ratio  of  this  difference 
to  the  given  7?-magnitudes.  Fechner  gave  this  law  a  precise 
phrasing  and  a  mathematical  derivation,  and  (as  we  indicated 
above,  pp.  xxiii  f.,  xxxviii)  put  it  to  elaborate  experimental  test. 


§14-     Demonstrations  of  Weber's  Law 


29 


Although  his  modesty  led  him  to  name  it  after  Weber,  we  might 
more  correctly  term  it  Fechner's  Law  or  the  Weber-Fechner 
Law. 


Generalised  representation  of  the  relation  between  S  and  R  formulated  in 
Weber's  Law.  Equal  sense-steps  are  marked  off  as  abscissas,  and  the  correspond- 
ing i?-values  are  entered  as  ordinates.     From  W.  Wundt,  Phys.  Psych.,  i.,  1902, 497. 

Our  object  in  the  following  experiments  is  to  demonstrate  or 
roughly  to  verify  some  of  the  results  which  are  summed  up  in 
Weber's  Law. 

The  formula  S  =  ^  log  R  can  be  worked  out,  though  not  quite  so  easily, 
in  terms  of  the  Z?Z,  of  j.  n.  sense-distances.  The  derivation  has,  however, 
only  a  mathematical  interest,  and  may  therefore  be  omitted  here. 

§  14.  Demonstrations  of  Weber's  Law. — The  object  of  the  fol- 
lowing tests  is  to  show  (i)  that  the  j.  n.  d.  of  brightness  is  rela- 
tively constant,  that  is,  that  the  difference  remains  j.  n.  with  very 
different  physical  intensities  of  the  component  Ry  provided  that 
these  physical  intensities  stand  always  in  the  same  ratio  to  each 
other;  and  (2)  that  a  series  of  equal  supraliminal  sense-distances 
is  paralleled  (approximately)  by  a  set  of  equal  7? -quotients. 

EXPERIMENT  IX 

Materials. — Spectacle  frame  (opticians'  trial  frame).  Three 
pairs  of  grey  ('  smoked ')  glasses,  of  different  shades,  rounded  to 
fit  the  frame.     A  partly  clouded  sky. 


30  Preliminary  Experiments 

Fechnef^s  Cloud  Experiment. — Find  a  place  in  the  sky  where 
the  junction  of  two  clouds,  or  of  two  parts  of  the  same  cloud, 
shows  to  the  naked  eye  just  a  trace  of  difference  in  brightness ; 
or  where  a  wisp  of  cloud  is  just  distinguishable  upon  the  back- 
ground of  sky.  Having  assured  yourself  (as  well  as  may  be,  in 
the  absence  of  objective  test)  that  the  brightness  difference  in 
question  is  really  only  just  noticeable,  proceed  to  observe  it  {a) 
through  the  lightest  grey  glasses,  {b)  through  the  middle  glasses^ 
{c)  through  the  darkest  glasses,  and  {d)  through  combinations  of 
the  glasses.  Make  the  observations  in  quick  succession,  in  order 
that  the  objective  illumination  of  the  sky  remain  constant  through- 
out the  experiment.  Does  the  apparent  brightness  difference 
suffer  any  change  under  the  changing  conditions  of  observation  > 

Now  reverse  the  experiment.  With  all  three  glasses  before 
the  eyes,  find  another  place  in  the  sky  which  shows  a  barely  per- 
ceptible brightness  difference.  Observe  this  in  turn  (a)  through 
the  two  darkest  pairs  of  glasses,  {b)  through  the  darkest  alone,. 
{c)  through  the  middle  pair  alone,  (</)  through  the  lightest,  and 
{e)  with  the  naked  eyes.  Does  the  apparent  difference  undergo 
any  change  1 

By  using  combinations  of  coloured  glasses,  one  may  reach  a  degree  of 
darkening  which  far  exceeds  that  given  by  the  smoked  glasses.  In  this  case^ 
however,  care  must  be  taken  that  the  two  cloud  shades,  or  the  brightnesses 
of  cloud  and  background,  are  pure  greys  and  show  no  trace  of  colour 
difference. 

Since  a  partly  overcast  sky  is  not  always  at  our  disposal,  we  must  have 
further  Materials  at  hand,  which  shall  render  us  independent  of  it.  The  ex- 
periment may  be  performed  in  various  other  ways. 

EXPERIMENT  IX  (a) 

Lay  out  upon  a  table,  on  an  uniform  black  or  white  background  and  in 
clear  diffuse  daylight,  a  set  of  Marbe's  grey  papers.  The  pieces  of  paper 
must  be  trimmed  accurately  to  the  same  size  (say,  4  by  i  cm.),  and  should 
be  covered  with  a  clean  strip  of  plate  glass  to  ensure  perfect  smoothness  of 
surface.  Select  two  papers  that  seem  to  the  naked  eye,  after  repeated  trial, 
to  be  j.  n.  d.  when  laid  edge  to  edge.  Observe  these  papers  through  the 
grey  glasses. 

With  all  three  glasses  before  the  eyes,  select  two  other  papers  that 
seem  to  be  j.  n.  d.  Observe  as  in  the  second  part  of  the  preceding  ex- 
periment. 

Repeat,  with  other  pairs  of  Marbe  papers,  as  often  as  the  series  allows. 


§  14.     Demonstrations  of  Weber's  Law  31 

EXPERIMENT  IX  (6) 

Place  upon  a  colour  mixer  two  greys  (mixtures  of  black  and  white  ;  large 
and  small  discs)  that  are  j.  n.  d.  Determine  the  required  i?-difference, 
roughly,  by  a  procedure  like  that  of  Experiment  I.  Observe  these  two 
greys,  in  diffuse  daylight,  through  the  grey  glasses.  Fixate  some  point 
upon  the  periphery  of  the  smaller  discs. 

With  all  three  glasses  before  the  eyes,  make  another  determination  of  a  j. 
n.  d.     Observe  as  in  the  second  part  of  the  cloud  experiment. 

Repeat  the  experiment,  with  at  least  two  other  determinations  of  a  j.  n. 
d.  under  both  conditions  (six  sets  of  observations  in  all). 

EXPERIMENT    X 

Materials. — Rumford  photometer,  with  screen  of  white  card- 
board. [The  photometer  consists  of  a  long  blackened  board, 
upon  which  are  pasted  converging  m.  scales.  A  rounded  pillar 
of  blackened  wood  (P,  Fig.  14)  is  set  at  the  centre  of  the  width 
of  the  board,  a  short  distance  from  one  end.  Behind  the  pillar 
and  at  the  edges  of  the  board  are  placed  two  upright  clips  of 
blackened  tin,  into  which  is  slipped  a  piece  of  white  cardboard,  S, 
which  serves  as  screen.]  Two  sources  of  light  {e.  g.,  a  standard 
candle  and  a  ground  glass  4  c.  p.  incandescent  lamp,  both  set 
upon  low  wooden  blocks),  with  curved  black  screens  of  cardboard 
or  tin.     Grey  glasses  and  frame,  as  before. 

Preliminaries. — The  photometer  is  laid  upon  a  low  table  in 
the  dark  room.  The  screens  of  the  lights  are  so  adjusted  that 
O,  who  stands  about  i  m.  behind  the  table  and  a  httle  to  one 
side,  can  see  nothing  of  candle  flame  or  lamp. 

When  the  lights  are  suitably  placed  upon  the  metre  scales,  the 
pillar  throws  two  shadows  on  the  white  screen.  We  are  con- 
cerned with  only  one  of  these  :  with  the  shadow  cast  by  the  can- 
dle. Let  L  and  C  represent  the  two  sources  of  light.  The 
shadow  due  to  C  is  illuminated  only  by  L  ;  the  rest  of  the  white 
screen  (apart  from  the  second  shadow,  which  does  not  interest 
us)  is  illuminated  by  both  L  and  C,  As  we  move  C  farther  and 
farther  back,  while  L  remains  in  position,  the  amount  of  light 
which  C  sends  to  the  screen  becomes  smaller  and  smaller,  until 
presently  it  is  imperceptible.  When  this  point  is  reached,  there 
is  no  difference  between  shadow  and  ground  ;  both  alike  are  (for 
sensation)  illuminated  only  by  Z.     A  very  shght  movement  of 


32 


Preliminary  Experiments 


C  forward  or  of  L  backward  will  now  suffice  to  bring  the  shadow 
into  view  again. 

Preliminary  experiments  are  to  be  made,  to  determine  posi- 
tions of  L  and  C  such  that  the  ^'-shadow  can 
easily  be  brought  to  disappearance.  ^    '""^ 

Fechner^s  Shadow  Experimeiit. — L  is  left 
in  its  place,  and  C  set  at  a  point  where  its 
shadow  is  clearly  seen  by  (9.  C  is  moved 
backward,  step  by  step, — O  closing  his  eyes 
between  observations, — until  the  shadow  dis- 
appears. Now  C  .is  moved  in,  slowly  and 
steadily,  until  the  shadow  becomes  j.  n.  C 
may  be  shifted  to  and  fro  about  this  point, 
as  O  desires,  until  a  positive  judgment  of  j.  n. 
can  be  passed.  No  single  observation  must 
be  long  continued,  or  after-images  will  arise. 

When  the  point  at  which  the  t7-shadow  be- 
comes j.  n.  has  been  finally  determined,   O 
makes    a    series    of  observations    with    the 
grey  glasses,  as    in    Exp.    IX.     Does  the 
shadow  and  ground  suffer  any  change } 

The  experiment  is  now  reversed.  With  all  three  glasses  before 
his  eyes,  O  determines  the  point  at  which  the  shadow  becomes 
j.  n.  Observations  are  then  taken,  as  in  the  second  part  of  Exp. 
IX.     Does  the  brightness  difference  show  any  change } 

Results. — E  has  a  full  record  of  (9's  introspections,  from  Exp. 
IX.^,  and  a  similar  record,  with  the  scale  values  of  the  photom- 
eter, from  Exp.  X.     The  following  Questions  arise. 

E  and  O  {\)  What  is  the  general  result  of  these  two  experi- 
ments }     What  does  it  prove  } 

O  (2)  Were  all  your  judgments  equally  certain  and  unhesi- 
tating }  If  not,  can  you  give  reasons  for  your  hesitation,  where 
it  occurred  }  Why  do  we  reverse  the  first  procedure  in  all  the 
experiments } 

6>  (3)  Are  you  satisfied  with  the  methods  employed  in  the  de- 
termination of  the  j.  n.  d. } 

1  Together  with  the  serial  numbers  of  the  Marbe  papers,  if  Exp.  IX.  (a)  was 
performed;  or  the  values  of  the  disc  sectors,  if  Exp.  IX.  {U)  was  done. 


Fig.  14. 
difference   between 


§  14-     Demonstrations  of  Weber's  Law        ,        33 

E  and  O  (4)  Can  you  suggest  any  variations  of  these  experi- 
ments that  may  serve  as  checks  upon  their  general  result,  or  will 
show  that  this  result  is  of  special  significance  ? 

E  and  O  {s)  Are  the  experiments,  as  performed,  rightly  called 
quantitative  ? 

E  and  O  (6)  Suggest  a  continuation  of  Exp.  X. 

E  and  O  (7)  Suggest  simple  alternative  experiments. 

EXPERIMENT   XI 

Materials. — Balance,  with  weights.  Set  of  weighted  envel- 
opes, 5  to  1 00  gr.  [The  envelopes  are  heavy  <  pay  '  envelopes,  of 
manilla  paper,  about  6.5  by  1 1  cm.  They  are  stiffened  by  a  piece 
of  cardboard  (light  for  the  lighter,  heavy  for  the  heavier  weights), 
cut  to  twice  the  size  of  the  envelopes  and  then  folded.  To 
weight  the  envelopes,  pieces  of  heavy  paper  or  of  card  are  pasted, 
or  strips  of  thin  sheet  lead  are  sewed,  to  the  inner  surfaces  of 
the  stiffening  card.  Scraps  of  loose  paper  may  be  dropped  into 
the  fold,  to  make  the  weight  exact.  The  weight  of  the  envelope 
in  gr,  is  written  on  the  inside  of  the  envelope  flap,  and  the  flap 
itself  tucked  in  ;  the  envelopes  are  not  to  be  sealed.  The  series 
consists  of  108  envelopes  :  {a)  extra  standard  of  5  gr. ;  {b)  extra 
standard  of  100  gr. ;  {c)  26  envelopes,  5  to  10  gr.,  differing  by 
0.2  gr.  increments;  (d)  30  envelopes,  10.5  to  25  gr.,  with  0.5 
gr.  differences  ;  (e)  25  envelopes,  26  to  50  gr.,  with  i  gr.  differ- 
ences ;  and  (/)  25  envelopes,  52  to  100  gr.,  with  2  gr.  differ- 
ences. ] 

Preliminaries. — O  sits  at  a  small,  low  table,  upon  which  the 
weighted  envelopes  are  laid.  The  two  extra  standards  are  placed 
apart ;  the  rest  are  piled  together  in  a  confused  heap. 

Sanford's  Weight  Experiment. — (9's  problem  is  to  arrange 
the  heap  of  envelopes  in  five  groups,  keeping  approximately 
equal  sense-distances  (intensive  differences)  between  group  and 
group.  The  envelopes  are  to  be  lifted  vertically,  being  taken  at 
the  flap  end  between  thumb  and  forefinger  of  the  right  hand. 
Reference  may  be  made,  as  often  as  desired,  to  the  extra  stand- 
ards, which  give  the  extreme  range  of  the  series ;  and  O  may  re- 
vise his  rating,  shift  envelopes  from  one  group  to  another,  etc., 
until  he  is  fully  satisfied  with  his  result. 
c 


34  Preliminary  Experiments 

Each  of  the  five  piles  is  then  weighed  in  the  balance,  and  the 
average  envelope-weight  in  each  pile  determined  by  division  of 
the  total  weight  by  the  number  of  envelopes.  Finally,  the  ratio 
of  these  average  weights  is  calculated  from  group  to  group. 

The  experiment  should  be  performed  at  least  twice  by  both 
E  and  O. 

Results. — The  numerical  results  (average  weight  of  the  en- 
velopes in  each  of  the  five  groups,  and  ratios  of  group  to  group) 
are  to  be  entered  in  the  note-book.  The  results  should  also  be 
expressed  graphically,  by  means  of  a  curve  :  the  group-numbers, 
I,  2,  3,  4,  5,  are  taken  as  abscissas,  and  the  average  weights  of 
the  envelopes  in  the  groups  as  ordinates. 

EXPERIMENT    XH 

Materials. — Set  of  Marbe  greys.  [This  is  a  series  of  44 
pieces  of  grey  paper,  ranging  in  brightness  from  dark  grey  to 
white.  A  convenient  size  is  3  by  6.5  cm.  The  pieces  are  num- 
bered on  the  back,  i  to  44,  in  the  order  light  to  dark.]  Sheet 
of  black  or  white  cardboard.  Sheet  of  plate  glass.  Kirschmann 
photometer. 

Preliminaries. — O  sits  at  a  low  table,  which  is  uniformly 
illuminated  by  diffuse  daylight.  The  cardboard  is  laid  on  the 
table,  and  the  44  papers  placed  in  order  upon  the  cardboard. 

EbbinghaiLs'  Brightness  Experiment. — (9's  problem  is  to  pick 
out  3  grey  papers  at  such  points  that  the  whole  series  of  greys 
is  divided  into  four  equal  sense-distances.  He  should  first  select 
the  middle  grey  of  the  series,  and  then  proceed  to  bisect  each  of 
the  half-series.  He  may  reconsider  his  choices  until  he  is  fully 
satisfied  with  his  result.  He  may  lay  the  five  terminal  greys 
(nos.  I,  44,  and  the  three  chosen)  by  themselves,  cover  them 
with  the  glass,  and  estimate  the  four  distances  apart  from  the 
intermediate  terms  ;  he  may  arrange  the  four  part-series  upon 
the  cardboard  one  above  another,  etc.,  etc.  The  glass  should 
always  be  used  before  final  choice  is  made.  When  he  has  de- 
cided, E  looks  at  the  backs  of  the  papers  and  records  their 
numbers. 

The  experiment  is  now  repeated,  with  reversal  of  the  order  of 
the  papers.     If  the  former  order  was  i  to  44,  from  left  to  right, 


§  14-     Demonstrations  of  Weber  s  Law  35 

it  is  now  I  to  44  from  right  to  left.  The  experiment  may  also 
be  repeated  upon  a  different  background  ;  or  the  papers  may  be 
given  to  (7  in  a  confused  heap. 

The  value  of  the  R  which  limited  the  equal  sense-distances  of 
Experiment  XI.  was  determined  by  help  of  the  balance.  To  de- 
termine the  values  of  our  five  grey  papers,  in  the  present 
experiment,  we  have  recourse  to  a  photometer. 

Kirschmann's  photometer  has  three  principal  parts,  (i)  The 
dark  chamber  is  formed  of  two  tubes  of  heavy  mill-board,  each 
50  cm.  in  length.  The  one,  open  at  both  ends,  is  20  cm.  in  in- 
side diameter ;  the  other,  closed  at  one  end,  is  just  large  enough 
to  slip  over  the  first.  Both  tubes  must  be  painted  and  repainted 
and  painted  again,  on  the  inside,  with  ivory  drop  black  or  some 
similar  dull  black  paint.  The  combined  tube,  varying  in 
length  according  to  circumstances  from  60  to  90  cm.,  is  sup- 
ported horizontally  upon  the  arms  of  two  solid  standards.  (2) 
Directly  in  front  of  the  dark  chamber  a  colour  mixer  is  set  up. 
It  is  necessary  that  the  rotating  discs  stand  close  up  to  the  dark 
chamber.  Hence  it  is  advisable  to  employ  a  small  electric  mixer, 
such  as  Zimmermann's,  which  is  supported  by  a  vertical  rod, 
vertically  adjustable.  A  semicircular  opening  is  cut  from  the 
lower  edge  of  the  front  tube ;  the  portion  of  the  vertical  rod 
which  projects  above  this  edge,  together  with  the  connecting 
wires,  is  carefully  blackened  ;  rod  and  wires  are  fitted  into  the 
opening,  so  that  the  discs  just  pass  the  edge  of  the  tube ;  the 
edges  of  the  semicircular  cut  are  blackened,  and  chinks  stopped 
by  bits  of  black  cloth.  (3)  The  discs  are  constructed,  as  a  rule, 
in  three  parts  ;  they  differ  somewhat,  under  different  conditions. 
Let  us  suppose  that  the  paper  which  we  are  to  test  is  a  black 
or  a  very  dark  grey.     The  discs  will  then  be  as  follows. 

T\i^ principal  disc  is  a  disc  of  white  cardboard,  21  cm.  in  di- 
ameter. Two  opposite  quadrants  are  left  entirely  white.  The 
other  two  are  left  white  for  a  space  of  2.5  cm.  from  the  centre, 
and  of  2  cm.  from  the  periphery.  The  remaining  spaces  of  4 
cm.  are  divided  into  two  quarter-rings  of  2  cm.  each.  The  inner 
quarter-rings  {a  in  Fig,  15,1)  are  covered  with  the  dark  grey 
paper  to  be   tested ;  the  outer  {b  of  the  Fig.)  are  cut  out,  so 


36 


Preliminary  Experiments 


that  the  black  of  the  dark  chamber  appears  through  them. 
The  edge  of  the  disc  carries  a  scale,  whose  units  are  0.5°.  The 
coarse  adjustment  (Fig.  15,2)  consists  of  a  disc,  9  cm.  in  di- 
ameter, of  the  same  white  cardboard ;  of 
a  white  index  sector,  extending  to  the  scale 
of  the  principal  disc  ;  and  of  a  quarter-ring, 
corresponding  to  those  of  the  previous  Fig., 
and  covered  with  the  same  dark  grey  paper. 
T\iQ,Jine  adjustment  (Fig.  1 5,3)  is  a  double 
sector  of  white  cardboard,  corresponding 
to  the  two  white  quadrants  of  the  principal 
disc.  For  convenience  of  manipulation,  it 
is  well  to  leave  a  tag  of,  say,  5''  in  width, 
projecting  some  2  mm.  from  the  two  ad- 
justment cards  beyond  the  periphery  of 
the  principal  disc. 

Let  the  apparatus  be  set  up,  the  discs 
so  arranged  that  a  and  b  of  the  principal 
disc  have  the  same  angular  magnitude. 
We  see  upon  the  white  ground,  as  the 
discs  are  rotated,  two  juxtaposed  grey 
rings,  the  inner  of  which  is  lighter  than 
the  outer.  These  rings  are  to  be  equated 
in  sensation.  We  therefore  add  more 
grey  to  the  inner  ring,  moving  the  coarse 
adjustment  by  means  of  the  index  sector. 
This  procedure  is  repeated  until  the  two 
greys  are  approximately  equal.  We  then 
make  use  of  the  fine  adjustment,  which 
enables  us  to  increase  (or  diminish)  the 
brightness  of  the  rings  by  equal  incre- 
ments of  white.  Let  us  assume,  e.  g.^  that  the  greys  are  ap- 
proximately equal  when  their  dark  components  are  of  60°  and 
62°  respectively.  If  we  move  the  fine  adjustment  back  through 
0.5°,  we  change  this  relation  to  61  :  63.  And  this  means  that  the 
ratio  of  the  absolute  black  to  the  paper  dark-grey  has  changed  by 

■7—r-  or The  fine  adjustment  must  be  varied  until  the  two 

62.63         1953  •  ■' 

grey  rings  merge  into  one,  without  the  slightest  trace  of  difference. 


Fig.    15. 

Discs   of    Kirschmann's 

photometer. 


§  14-     Demonstrations  of  Weber  s  Law  37^ 

At  the  conclusion  of  the  experiment,  we  have  the  values  ^° 
and  b°  of  the  paper  grey  and  the  chamber  black.  If  we  put  the 
reflecting  power  of  the  white  cardboard  =  i,  that  ot  the  dark 
grey  paper  =  x^  and  that  of  the  chamber  black  =  o,  we  have 
the  equation : 

(360  —  a)^ax=  (360  —  b), 

or  X  =  :?=_*. 

a 

It  is  now  a  simple  matter  to  determine  the  brightness  values  of 
lighter  greys.  We  may  retain  the  dark  chamber,  reducing  the 
angular  value  of  the  b  of  the  principal  disc,  and  increasing  the 
values  of  a  in  it  and  in  the  coarse  adjustment ;  or  we  may  do 
away  with  the  chamber,  and  construct  a  principal  disc  whose  b 
consist  of  our  known  dark  grey  paper  and  whose  a  are  covered 
by  the  lighter  paper  to  be  tested. 

Results. — E  has,  we  will  suppose,  the  numbers  of  the  papers 
chosen  by  O  in  the  two  space  orders  of  the  experiment ;  he  has 
also  determined  the  photometric  values  of  the  papers  in  terms  of 
the  white  cardboard  of  the  photometer.  All  these  results,  with 
sample  pieces  of  the  papers,  are  to  be  entered  in  the  note-book. 
The  photometric  values  of  the  two  choices  are  now  averaged ; 
the  averages,  together  with  their  ratios,  are  entered  in  the  note- 
book ;  and  a  curve  is  drawn  like  that  of  Experiment  XI.  If 
time  allows  of  the  repetition  or  variation  of  the  experiment,  the 
additional  results  are  recorded  in  the  same  way. 

Questions. — E  and  O  (8)  What  is  the  general  result  of  these 
two  experiments  ?     What  does  it  prove } 

E  and  O  (9)  What  are  the  principal  sources  of  error,  object- 
ive and  subjective,  in  the  experiments  as  performed  ? 

^  and  O  {\o)  Suggest  further  experiments  of  the  same  kind. 

£■  and  6^  (11)  Suggest  a  more  methodical  and  reliable  way 
of  securing  the  result  aimed  at  in  these  experiments. 


CHAPTER  II 

THE    METRIC    METHODS 

§15.  The  Law  of  Error. — Suppose  that  you  are  required  to 
measure  the  length  of  a  rectangular  block  of  wood,  and  that  you  are 
given  a  mm.  scale  to  measure  with.  Carefully  adjusting  the  scale, 
and  estimating  to  tenths  of  your  unit,  you  find  that  the  length  of 
the  block  is  1 1 2.9  mm.  In  reporting  this  measurement,  you  have 
no  doubt  that  your  result  is  accurate  to  i  mm.;  you  feel  rea- 
sonably sure  that  it  is  accurate  to  0.5  mm.  But  you  are  ready 
to  admit  that  you  may,  possibly,  have  made  an  error  of  0.25 
mm.;  you  think  it  very  likely  indeed  that  you  are  wrong  by  o.i 
mm.  In  the  interests  of  accuracy,  then,  you  are  tempted  to 
verify  your  first  measurement  by  another. 

Your  second  measurement  may  repeat  the  result  of  the  first. 
More  probably  it  will  not.  But  if  it  does,  you  will  not  be  justi- 
fied in  accepting  the  1 12.9  mm.  as  the  true  value  of  the  length 
you  are  measuring.  For  if  you  repeat  the  measurement,  not 
once  only,  but  a  great  many  times,  you  will  find  that  the  results 
vary  in  a  somewhat  surprising  way.  Thus  10  successive  meas- 
urements, though  made  with  the  same  scale  and  apparently 
under  the  same  conditions  of  observation,  might  give  you  the 
values:  11 2.9,  11 3.0,  11 2.9,  11 2.9,  11 2.9,  11 2.9,  11 3.1,  11 2.9, 
1 1 3.0,  1 1 3. 1  mm.  It  seems  natural,  perhaps,  to  lay  these  dis- 
crepancies to  the  account  of  the  measuring  instrument :  your 
unit  is  I  mm.,  and  the  tenths  are  estimated.  But  the  same  thing 
would  happen  if  you  had  a  scale  that  read,  with  mechanical  accu- 
racy, to  0.00 1  mm.  In  this  case,  the  actual  magnitude  of  the 
discrepancies  would,  of  course,  be  reduced.  But  the  discrepan- 
cies themselves  would  still  remain ;  and,  indeed,  relatively  to  the 
size  of  the  unit  would  be  larger  than  before, — since,  where  the 
unit  of  measurement  is  very  small,  the  numerical  measure  of 
discrepancies  is  naturally  increased.     If  you  found  differences  of 

38 


§  15-     The  Law  of  Error  39 

I  or  2  tenths  before,  you  will  now  very  possibly  find  differences 
of  several  thousandths. 

These  facts  mean,  in  brief,  that  every  observation  you  make 
is  subject  to  an  error:  the  error  being  defined  as  the  difference 
(i)  between  the  observed  and  the  true  value,  if  the  true  value 
is  known,  or  (2)  between  the  observed  and  the  most  probable 
value,  if  the  true  value  is  unknown.  Provided  that  there  is  no  bias, 
no  constant  pressure  upon  you  to  err  more  often  in  one  direction 
than  in  another,  the  *  most  probable  value '  is,  evidently,  the  arith- 
metical mean  or  average  of  all  the  observed  values.  The  errors 
are  then  determined  as  the  differences  between  this  mean  and  the 
single  measurements.  The  numerical  values  represent  their 
magnitude ;  the  sign  prefixed  to  them,  plus  or  minuSj  represents 
their  direction. 

In  the  instance  taken,  of  measuring  the  wooden  block,  there  seems  to  be 
no  room  for  an  error  of  bias.  Nevertheless,  it  will  be  advisable  to  vary 
your  conditions  :  to  lay  the  o-point  of  your  scale  as  often  upon  the  right  as 
upon  the  left  edge  of  the  block,  and  so  to  measure  from  right  to  left  as  often 
as  you  measure  from  left  to  right ;  and,  again,  to  turn  the  block  round,  so 
that  its  original  right  becomes  left,  and  conversely,  and  to  execute  both 
sets  of  measurements  in  both  positions  of  the  block.  The  task  is  so  simple, 
and  the  change  of  conditions  so  slight,  that  the  various  sets  of  measure- 
ments will,  in  all  probability,  be  interchangeable.  In  this  event,  the  arith- 
metical mean  of  all  measurements  gives,  as  we  have  said,  the  most  proba- 
ble length  of  the  block.  If,  on  the  other  hand,  there  is  any  evidence  of 
what  are  called  '  constant '  or  '  systematic  errors,'  these  must  first  be  elimi- 
nated,— an  easy  matter,  but  one  which  we  cannot  here  discuss, — and  the 
corrected  mean  employed  for  the  determination  of  the  errors  of  observation. 

Mathematicians  regard  these  *  errors  of  observation  '  as  the  al- 
gebraical sum  of  a  very  large  number  of  elemental  errors,  due  to 
various  unknown  causes.  If  the  cause  of  an  error  is  known,  its 
result  really  ceases  to  be  an  error,  in  the  present  meaning  of  the 
word  ;  for,  knowing  the  cause,  you  can  modify  your  procedure  to 
counteract  it,  and  can  thus  eliminate  its  result.  The  errors  of 
observation,  which  produce  the  discrepancies  in  your  measure- 
ments, are  accidental  errors,  due  to  unknown  conditions. 

It  does  not  follow,  however,  that  because  we  know  nothing  of 
the  causes  of  a  set  of  errors  we  can  say  nothing  about  the  errors 
themselves.     On  the  contrary,  mathematical  theory  has  a  good 


40  The  Metric  Methods 

deal  to  say  about  the  errors  of  observation  ;  and  part  of  what  it 
has  to  say  depends — paradoxical  as  the  statement  may  sound — 
upon  this  very  fact  of  our  ignorance  of  the  causes  of  error.  The 
theory  tells  us,  for  instance,  (i)  that  smaller  errors  are  more 
likely  to  occur  than  larger.  This  canon  is,  undoubtedly,  sug- 
gested by  experience ;  we  do  not  make  errors  of  i  cm., — at  least, 
the  accurate  observer  does  not, — in  measuring  with  a  mm. 
scale.  It  is  also  a  direct  consequence  of  the  mathematical  way 
of  regarding  the  errors  :  for  the  elemental  errors,  of  which  the 
resultant  errors  of  observation  are  made  up,  may  have  either  di- 
rection, and  therefore  tend  to  cancel  one  another  ;  hence  small 
errors  must  be  more  frequent  than  large,  and  the  smallest  error 
— the  error  o — the  most  frequent  of  all.  The  theory  tells  us 
further,  (2)  that,  in  the  long  run,  phis  and  minus  errors  will 
occur  with  the  same  frequency.  This  may  certainly  be  assumed, 
just  because  we  are  supposed  to  know  nothing  of  the  causes 
of  the  errors ;  and  it  is  verified  by  actual  results.  The 
theory  tells  us,  finally,  (3)  that,  while  smaller  errors  are  more 
frequent  than  larger,  the  errors  as  a  whole  may  be  regarded  as 
forming  a  continuously  graded  series,  ranging  from  o  to  00  .  This 
continuous  gradation  cannot  be  observed  in  practice  ;  if  observa- 
tions were  repeated  indefinitely  upon  the  same  magnitude  and 
under  the  same  circumstances,  only  a  limited  number  of  observed 
values,  all  of  them  exact  multiples  of  the  unit  of  the  instrument,^ 
would  occur.  What  the  theory  does,  therefore,  is  to  regard  the  unit 
of  the  instrument  as  reduced  without  limit.  This  convention 
renders  a  mathematical  treatment  of  results  very  much  easier 
than  it  would  otherwise  be  ;  we  need  not  hesitate  to  adopt  it. 
We  shall,  perhaps,  hesitate  to  accept  the  statement  that,  even 
in  an  infinitely  long  series  of  observations,  there  must  occur  an 
error  of  infinite  magnitude  ;  but  we  may  comfort  ourselves  by 
the  thought  that  it  is  considered  to  occur  infinitely  seldom.  Or 
we  may,  if  we  like,  say  that  the  errors  of  observation  form  a  con- 
tinuously graded  series,  ranging  between  o  and  some  fixed  upper 
limit  of  magnitude  :  the  extension  of  this  limit  to  00  is  made  sim- 
ply for  mathematical  reasons,  and  has  no  practical  significance. 
These  three  rules,  we  notice,  are  all  concerned  with  the  rela- 

lOr,  as  in  the  case  of  estimated  tenths,  aUquot  parts  of  the  unit. 


§  15.      The  Law  of  Error  41 

tion  of  the  magnitude  of  an  error  to  \X.^  frequency.  Small  errors 
are  relatively  more  frequent  than  large  ;  plus  and  minus  errors 
of  the  same  magnitude  are  of  equal  relative  frequency  ;  all  errors, 
of  whatever  magnitude,  have  a  certain  frequency  of  occurrence. 
It  is  customary  to  express  such  a  relation  between  two  magni- 
tudes in  graphic  form,  by  means  of  a  curve.  The  curve  is,  then, 
a  line  whose .  course  shows  us,  at  a  glance,  the  relation  in  which 
our  two  variable  quantities  (here,  magnitude  and  frequency  of 
error)  stand  to  each  other. 

To  construct  a  curve,  we  first  draw  two  straight  lines  XX'  and 
YY'  at  right  angles  to  each  other,  causing  them  to  intersect  at 
the  point  O.  The  lines  XX'  and  YY'  are  called  the  axes  of  co- 
ordinates,^ and  the  point  O  the  origin  of  coordinates.  The  curve 
is  given,  if  we  know  the  relation  between  the  two  *  coordinates  *' 
of  any  point  upon  it.  Let  the  point  P  be  given.  Its  distance 
from  the  axis  YY'  is  NP  or  OM ;  this  distance  is  called  x,  the 
abscissa  of  P.  Its  distance  from  the  axis  XX'  is  ON  or  MP ;  this- 
distance  is  called  j,  the  ordinate  of  P.  The  abscissa  and  the 
ordinate  are  together  called  the  coordinates  of  P.  The  rela- 
tion of  X  to  J,  in  Fig.  1 6,  is  such  that  y  =  log  x :  this  is  the 
*  equation  '  of  this  particular  curve.  Knowing  it,  we  can  describe 
the  complete  curve,  or  as  much  of  it  as  we  require,  as  is  shown 
in  the  Fig.^  We  do  this  by  assigning  successive  values  to  the 
independent  variable  x ;  computing  the  corresponding  values  of 
the  dependent  variable  J  ;  then  laying  down  the  successive  points 
whose  coordinates"  are  {x^y) ;  and  finally  drawing  a  smooth  curve 
through  them. 

If  we  apply  these  rules  of  graphic  representation  to  our  errors 
of  observation,  we  obtain  a  bell-shaped,  symmetrical  curve  of  the 
kind  shown  in  Fig.  17.  The  point  O  upon  XX'  corresponds  to 
an  error  of  the  value  o  ;  the  magnitude  of  error,  positive  or  neg- 
ative, is  measured  along  XX',  to  right  and  left  of  O ;  the  rela- 
tive frequency  of  an  error  of  a  given  magnitude  is  shown  by  the 

1  The  line  XX'  or  XOX'  is  called  the  axis  of  x,  the  line  YY'  or  YOY'  the  axis 
oi'  y. 

3  W^e  term  x  positive  when  it  is  to  the  right  of  O,  and  negative  when  it  is  to  the 
left ;  we  term  y  positive  when  it  is  above  O,  and  negative  when  it  is  below.  This, 
arrangement  oi  plus  and  minus  values  is,  of  course,  conventional. 


42 


The  Metric  Methods 


height  of  the  corresponding  ordinate.  Thus,  an  error  of  the 
magnitude  o  occurs  most  often ;  an  error  of  the  magnitude  OM 
occurs  with  the  relative  frequency  ON. 

Let  us,  now,  examine  the  curve  in  some  little  detail.  We  no- 
tice, first,  that  the  curve  cuts  the  axis  of  y  perpendicularly ;  in 
other  words,  the  curve  in  that  neighbourhood  is  nearly  horizontal : 
this  implies  that,  in  the  neighbourhood  of  the  arithmetical  mean, 
there  is  a  group  of  values  that  are  approximately  equal.  After 
a  time,  the  curve  begins  to  slope  away  rapidly  towards  the  axis 
of  X :  this  implies  that  the  observed  values  begin  to  get  less  fre- 
quent as  we  depart  from  the  mean ;  in  other  words,  that  larger 
errors  are  less  frequent  than  smaller.     The  curve  approaches  the 


—Xz 


T 

Fig.  i6. 
From  Cattell,  art.  Curve,  in  Baldwin's  Diet,  of  Phil,  and  Psych.,  i.,  1901,  249. 
If  YY'  be  laid  off  in  units  of  sensation,  and  XX'  in  units  of  stimulus,  the  equation 
y  =z  log  X  becomes,  specifically,  s  =  log  r,  Fechner's  formula  for  Weber's  Law. 

axis  of  X,  in  both  directions,  asymptotically  ;  that  is,  it  comes  in 
course  of  time  indefinitely  close  to  that  axis,  but  never  actually 
touches  it  :  this  implies  that  no  magnitude,  however  remote  from 
the  mean,  is  strictly  impossible ;  every  error,  however  excessive, 


§  15-      The  Law  of  Error  43 

will  have  to  occur  at  length,  within  the  range  of  a  sufficiently 
long  experience.  It  should,  however,  be  said  that  no  graphic 
representation  can  give  an  adequate  idea  of  the  extreme  rapidity 
with  which  the  curve  tends  toward  the  axis  of  x.  To  the  eye, 
the  two  appear^  after  a  very  short  course,  to  merge  into  each 
other.  This  shows  how  small  is  the  chance  of  even  a  moderate 
error.  Finally,  the  curve  is  symmetrical  about  the  axis  of  y : 
this  implies  that  equal  deviations  from  the  mean,  in  excess  and 
defect,  tend  in  the  long  run  to  appear  with  equal  frequency. 

The  equation  of  this  error  curve  is  not  quite  so  simple  as  the 
equation  of  the  logarithmic  curve  given  above.  It  is,  however, 
an  equation  of  the  same  kind.  Both  alike  are  *  exponential ' 
equations :  that  is,  equations  in  which  one  of  the  two  variables 
(x,  y)  appears  only  as  the  exponent  of  some  other  quantity.  In- 
stead of  taking  the  form  y=^f  {x), — ^where  /  is  the  general  sign 


.  Fig.  17. 

Curves  of  Gauss'  Law  of  Error,  for  observations  of  less  and  greater  accuracy. 

of  function,  or  functional  symbol,  and  the  equation  reads  **jj/ 
varies  in  some  determinate  way  whenever  x  varies," — the  expo- 
nential equation  takes  the  form  y=f  (^),  where  c  is  some  con- 
stant number,  and  we  read  "  j  varies  in  some  determinate  way 
whenever  a  certain  constant  quantity  raised  to  the  ;ir-power 
varies."  We  cannot  here  follow  the  mathematical  reasoning 
which  leads  to  this  exponential  form  of  equation.  We  can,  how- 
ever, decide,  in  the  light  of  a  principle  already  laid  down,  what 
the  nature  of  the  ;r-exponent  in  the  equation  of  the  error  curve 
shall  be.  We  have  said  that  positive  and  negative  errors  will,  in 
the  long  run,  appear  with  equal  frequency.  But,  in  saying  this, 
we  have  said  that  the  frequency  of  error  must  be  a  function  of 


44  The  Metric  Methods 

an  even  power  of  the  magnitude  :  that  is,  y  must  be  a  function 
of  x^y  or  x^^  or  ;r^,  etc. :  otherwise  the  frequency  of  the  same 
amount  of  error  would  vary  according  as  the  error  were  positive 
or  negative.  The  even  powers  are  always  intrinsically  positive, 
whether  x  itself  be  positive  or  negative.  We  shall  therefore 
find,  in  our  exponential  equation,  not  x^  but  x^  or  x^  or  what  not. 
The  use  of  x^  rather  than  x^  leads  to  formulae  of  comparative 
simplicity,  which  can,  fortunately,  be  employed  without  fear  of 
error.  1 

The  equation  of  the  error  curve  is,  then,  an  exponential  equa- 
tion in  which  the  magnitude  of  error  x  appears  in  the  form  x^. 
The  equation  is  usually  written  : 

h      —K^xi- 

The  symbols  are  by  no  means  so  formidable  as  they  look.  The 
values  77  and  e  are  numerical  constants  :  it  is  3.14,  the  ratio  of 
the  circumference  to  the  diameter  of  a  circle,  and  e  is  2.72,  the 
base  of  the  system  of  natural  logarithms.  However  they  may 
have  got  into  the  equation,  then,  we  can  express  them  in  ordi- 
nary numbers.     So  the  equation  becomes  : 

y  =  0.56  //.   (2.72)— ^^2x2^      or 
y  ^       056 /^ 
(2.72)^2-^2 

The  only  unfamiliar  quantity  in  this  equation  is  now  the  con- 
stant //.  This  is  a  value  which,  while  constant  for  any  given  set 
of  observations,  depends  upon  the  circumstances  under  which 
the  observations  are  made.  It  is  a  direct  measure  of  the  ob- 
server's accuracy,  of  the  precision  of  his  measurements.  It  is 
thus  a  value  akin  to  the  MV;  only  that  the  MV  measures  the 
observer's  accuracy  inversely, — the  larger  the  MV,  the  smaller 
the  accuracy,  and  conversely, — while  h  measures  it  directly. 
Where  h  is  large,  the  error  curve  will  be  tall  and  narrow ;  the 
observed  values  group  closely  about  the  mean ;  small  errors  are 
very  greatly  in  preponderance.  Where  h  is  small,  the  curve  is 
low  and  broad ;  the  errors  are  more  evenly  distributed  for  a  con- 

1  The  argument  may  be  made  clearer,  as  follows.  The  requirement  is  that 
y^/ (c(-x)*^)  =y(r (+«)«),  If  fi  is  I,  3,  5,  .  .  the  requirement  cannot  be  ful- 
filled; if  «is  2,  4,  6,  .   .  the  equation  holds. 


15.      The  Lazv  of  Error  45 


siderable  distance  on  either  side  of  the  mean ;  large  errors  are 
relatively  common.  The  two  curves  of  Fig.  1 7  are,  both  alike, 
curves  of  error ;  both  satisfy  the  equation  which  we  have  just 
discussed.  But  the  left-hand  curve  has  a  small  //,  the  right-hand 
a  large  //,  in  its  equation.  It  is  h  that  determines  the  special 
form  which  the  bell-shaped  curve  assumes,  as  the  graphic  repre- 
sentation of  the  errors  made  in  a  particular  series  of  observations. 

If  we  translate  our  equation  into  words,  it  will  read  somewhat 
as  follows.  The  relative  frequency  of  a  given  error  (j/),  in  any 
extended  set  of  observations,  is  equal  to  a  certain  constant  value, 
divided  by  another  constant  value  raised  to  the  second  power  of 
the  product  of  the  number  representing  the  magnitude  of  error 
{x)  and  a  third  constant  value.  The  constants  of  numerator  and 
denominator  are  not  independent  values ;  both  ahke  are  deter- 
mined by  the  observer's  accuracy  (both  contain  //),  and  there- 
fore measure  his  precision.  If  we  know  the  x  and  y  of  any  point 
upon  the  curve  ;  if  we  know,  that  is,  the  relative  frequency  of  any 
single  error ;  we  can  find  the  value  of  h^  and  so  construct  the 
whole  error  curve. 

At  this  point,  we  are  ready  to  discuss  a  new  term, — the  term 
*■  chance '  or  (what  is  mathematically  the  same  thing)  *  probability.' 
We  have  already  spoken  of  the  arithmetical  mean  as  the  *most 
probable '  value  yielded  by  a  series  of  observations.  Let  us  now 
see  precisely  what  probability  means. 

The  probability  of  a  given  event,  the  chance  of  its  occurrence, 
is  defined  as  the  numerical  fraction  which  represents  the  propor- 
tion holding,  in  the  long  run,  between  this  particular  event  and 
the  indefinitely  large  series  of  similar  events  within  which  it  falls. 
A  few  illustrations  will  make  the  definition  clear.  ( i )  What,  for 
instance,  is  the  probability  that  a  certain  infant  will  live  to  be  80 
years  old }  The  particular  event  here  is  a  life  from  infancy  to 
80  years ;  the  indefinitely  large  series  of  similar  events  comprises 
the  life-terms  of  all  infants.  Let  the  proportion  of  the  former 
to  the  latter  be  found,  by  statistical  investigation,  to  be  1:9;  in 
other  words,  suppose  that  i  infant  in  10  does,  as  a  matter  of  fact, 
attain  the  age  of  80.  Then  the  chance  or  probability  that  our 
particular  infant  will  reach  that  age  is  defined  by  the  numerical 


46 


The  Metric  Methods 


fraction^.  Again,  (2)  what  is  the  probability  that  a  player 
will  throw  a  3  in  a  given  cast  of  a  die  ?  The  die  has  6  faces ; 
and  the  probability  that  he  throws  any  one  of  them  is  defined 
by  the  numerical  fraction  \.     This  does  not  mean,  of   course, 

Mechanical  Illustration  of  the  Cause  of  the  Curve  of 
Frequency:  from  F.  Galton,  Natural  Inheritance,  1889, 
63. — "A  frame  is  glazed  in  front,  leaving  a  depth  of  about 
a  quarter  of  an  inch  behind  the  glass.  Strips  are  placed 
in  the  upper  part  to  act  as  a  funnel.  Below  the  outlet  of 
the  funnel  stand  a  succession  of  rows  of  pins  stuck  squarely 
into  the  backboard,  and  below  these  again  are  a  series  of 
vertical  compartments.  A  charge  of  small  shot  is  en- 
closed. .  .  .  The  shot  passes  through  the  funnel  and, 
issuing  from  its  narrow  end,  scampers  deviously  down 
through  the  pins  in  a  curious  and  interesting  way ;  each 
of  them  darting  a  step  to  the  right  or  left,  as  the  case 
may  be,  every  time  it  strikes  a  pin.  The  pins  are  dis- 
posed in  a  quincunx  fashion,  so  that  every  descending 
shot  strikes  against  a  pin  in  every  successive  row.  The 
cascade  issuing  from  the  funnel  broadens  as  it  descends, 
and,  at  length,  every  shot  finds  itself  caught  in  a  compart^ 
ment  immediately  after  freeing  itself  from  the  last  row 
of  pins.  The  outline  of  the  columns  of  shot  that  ac- 
cumulate in  the  successive  compartments  approximates 
to  the  Curve  of  Frequency,  and  is  closely  of  the  same 
shape,  however  often  the  experiment  is  repeated." 


Fig.    18. 


that  in  6  successive  throws  he  will  get  the  6  faces,  i,  2,  3,  4,  5,  6, 
once  each.  It  means  that,  in  an  indefinitely  long  series  of  throws, 
any  particular  face  will  turn  up  as  often  as  any  other.  Since 
there  are  6  faces  in  all,  the  3-face  must  appear  in  J  of  all  the 
throws  made.  Once  more,  (3)  suppose  that  a  friend  of  mine  sails 
on  a  boat  with  19  other  passengers,  and  that  I  receive  the  news 
of  the  loss  of  a  passenger  during  the  voyage.  What  is  the  prob- 
ability that  the  lost  passenger  will  prove  to  be  my  friend  >  It  is 
defined  by  the  fraction  gV-  ^^^^  ^o^s>  not  mean,  again,  that 
if  my  friend  planned  to  take  20  successive  journeys,  under  pre- 
cisely the  same  circumstances,  he  would  be  planning  his  death, — 
for  the  reason  that  he  would  certainly  be  lost  on  one  of  them. 
It  means  rather  that,  if  we  had  data  from  an  indefinitely  long 
series  of  voyages,  undertaken  always  by  20  passengers  carrying 
the  badges  A,  B,  C,  .  .  .  ,  and  resulting  always  in  the  loss  of 
one  man,  the  carrier  of  A — which  we  will  assume  to  the  badge 


§  15-      The  Law  of  Error  47 

of  friendship — would  be  lost  as  often  as  the  carrier  of  B,  C,  etc.: 
that  is,  in  -^  of  the  whole  number  of  cases. 

In  brief,  to  say  that  the  ^chance  of  a  given  event's  happening 
in  a  certain  way  is  ^  or  y^,  or  what  not,  is  only  another 
way  of  saying  that,  in  the  long  run,  it  does  tend  to  happen  in  that 
way  once  in  5  or  1 00  or  so  many  times.  Now  we  have  seen  that 
the  7  of  the  equation  of  the  error  curve  represents  the  relative 
frequency  of  occurrence  of  a  given  error  in  an  indefinitely  ex- 
tended set  of  observations.  This  means,  then,  that  the  y  repre- 
sents the  probability  of  the  x  with  which  it  is  correlated.  If  we 
denote  the  probability  of  occurrence  of  an  error  Xy  of  definite 
magnitude,  by  the  symbol  /*^,  we  may  substitute  P^  for  y  in  the 

equation,  and  write:    P,^  =  ^J—^~^^'^^'^ .     Since,  again,  the  most 

probable  of  the  observed  values  is  the  value  of  the  arithmetical 
mean,  and  the  error  of  the  arithmetical  mean  is  o,  it  follows  that 
the  probability  of  the  most  probable  error  (the  height  of   the 

highest  ordinate  of  the  error  curve)  is  P(i  =  -zz.^ 

If  we  could  rest  here,  our  discussion  would  be  comparatively^ 
simple.  But  we  cannot.  We  must,  indeed,  retrace  our  steps  a 
little.  For,  in  regarding  the  equation  of  the  error  curve  as  an 
ordinary  algebraical  equation,  we  have  left  out  of  account  the 
fact  that  the  errors  are  supposed  to  be  continuously  graded,  that 
the  ordinates  of  the  curve  are  infinite  in  number.  We  have  said 
that  y  *■  represents  '  the  relative  frequency  of  an  error  of  the  mag- 
nitude Xy  or  the  probability  of  the  x  with  which  it  is  correlated. 
Such  a  statement  is  not  strictly  correct ;  we  should  have  said 
that  y  is  proportional  to  the  frequency  of  x.  Since  the  possibil- 
ity of  error  is  continuous,  the  actual  number  of  errors  of  a  par- 
ticular magnitude  must,  of  course,  be  indefinitely  small.  Since 
an  infinite  number  of  ordinates  may  be  drawn,  the  occurrence  of 
any  one  definite  amount  of  error  is  infinitely  improbable.  What> 
then,  becomes  of  our  derived  equations  ? 

Let  us  remind  ourselves,  first  of  all,  that  the  unit  of  measure 
ment  in  laboratory  work  is  not  a  point  but  a  distance.     We  refer 

1  Because,  when  x  =  o,  the  symbol  ^--^^-^^  becomes  ^,  and  <?»  =  i.  This  is  a 
direct  consequence  of  the  law  or  convention  that  a^  y^  av  =  a^"^  v.  Let  ^  =  o  r 
then  a^  X  av  =  a*,  i.e.,  aV  =  i. 


48 


The  Metric  Methods 


all  our  measurements,  it  is  true,  to  a  scale  mark ;  but,  in  doing 
this,  we  refer  to  the  mark  all  the  measurements  that  fall  within 
certain  hmits.     We  read  to  the  neairsi  mm.,  or  o.  i  mm.,  or  what 


• 

_ 

8 

•.'•' 

7 

s 

- 

6 

5 

^j^i 

- 

4 

■'■;■ 

3 

- 

2 

1 

. 

w 

■ 

■ 

Fig.  19, 


Illustration  of  a  Normal  Distribution: 
from  F.  Galton,  Hereditary  Genius,  1892, 
24. — "Suppose  a  million  of  the  men  [of 
a  supposed  island  race,  freely  intermarry- 
ing, and  living  for  many  generations  un- 
der constant  conditions]  to  stand  in  turns, 
with  their  backs  against  a  vertical  board 
of  sufficient  height,  and  their  heights  to 
be  dotted  off  upon  it.  The  board  would 
then  present  the  appearance  shown  in  the 
diagram.  The  line  of  average  height  is 
that  which  divides  the  dots  into  two  equal 
parts,  and  stands,  in  the  case  we  have 
assumed,  at  the  height  of  66  in.  The 
dots  will  be  found  to  be  arranged  so  sys- 
tematically on  either  side  of  the  line  of 
average,  that  the  lower  half  of  the  dia- 
gram will  be  almost  a  precise  reflection  of 
the  upper.  Next,  let  100  dots  be  counted 
from  above  downwards,  and  let  a  line  be 
drawn  below  them.  According  to  the 
conditions,  the  line  vnll  stand  at  the 
height  of  76  in.  Using  the  data  afforded 
by  those  two  lines,  it  is  possible,  by  the 
help  of  the  law  of  deviation  from  an  aver- 
age, to  reproduce,  with  extraordinary 
closeness,  the  entire  system  of  dots  on 
the  board." 


not.  If,  then,  we  wish  to  pass  from  the  continuous  to  the  dis- 
crete, from  the  infinitely  numerous  ordinates  of  the  error  curve 
to  the  detached  ordinates  that  we  measure  in  practice,  the  cru- 
cial question  is,  not  *  What  is  the  relative  frequency  j/  of  a  defi- 
nite amount  of  error  x  } '  but  rather  '  What  is  the  relative  fre- 
quency of  the  errors  falling  between  two  consecutive  divisions  of 
our  scale  .? '  This  is  given,  not  simply  by  the  height  of  the  ordi- 
nate _;;/  correlated  with  a  determinate  x,  but  rather,  as  we  shall 
see  in  a  moment,  by  the  area  of  that  portion  of  the  curve  com- 
prised between  the  ordinates  jy  of  the  abscissas  xx'.  To  ascer- 
tain it,  we  must  sum  up  all  the  ordinates  of  the  curve  (infinite  in 
number)  between  the  prescribed  limits  ;  we  must  then  reduce  this 


§  15.      The  Law  of  Error  49 

sum  to  a  proportional  finite  value  ;  and  this  finite  value  will  rep- 
resent the  probability  of  an  error  within  the  scale  unit  xx' , 

Have  we  mended  matters  ?  Can  we  sum  a  portion  of  an  infi- 
nite series  to  a  finite  result  ?  Yes  ! — but  only  by  the  use  of  the 
calculus.  Suppose  that  we  have  two  quantities,  like;ir  and  jj/,  that 
are  continuously  variable.  Suppose,  again,  as  is  the  case  with  x 
and  J,  that  we  have  an  equation  involving  the  two  quantities. 
The  quantities  will,  in  consequence  of  the  equation,  vary  together, 
so  that  there  will  be  equations  between  their  rates  of  change. 
Now  let  dx  or  dy  stand  for  an  infinitely  small  increment  (a  *  dif- 
ferential') of  ;i;  ox  y.  The  differential  calculus  treats  of  the  ra- 
tios of  these  differentials,  and  of  the  fundamental  formulae  into 
which  the  ratios  enter ;  the  integral  calculus  treats  of  the  sum- 
mation (integration)  of  an  infinite  series  of  differentials  to  a  fin- 
ite value.  Here,  then,  in  the  calculus,  is  the  very  instrument  of 
which  we  are  in  search. 

We  write  our  revised  formula  as  follows.  P^^  which  we  em- 
ployed for  the  probability  of  Xy  comes  to  mean  the  relative  fre- 
quency of  errors  within  the  minimal  interval  x  X.ox  -\-  dx.  We 
assume  that  the  ordinate  y  is  constant  throughout  this  small  in- 
terval ;  so  that  P^  must  now  evidently  contain  dx  as  a  factor. 
We  have  accordingly : 

P  =  ^-e-h'^^-'dx. 

^^ 
No  objection  can  be  taken  to  our  assumption  of  the  constancy 

of  y,  if  we  sum  up  an  infinite  number  of  such  terms,  and  proceed 

to  the  limit  where  dx  is  made  indefinitely  small.  ^     Doing  this, 

*  The  above  argument  will,  perhaps,  be  rendered  clearer  if  we  take  a  geometri- 
cal illustration.  Consider  any  portion  MN oi  a  curve  whose  equation  is  j  r^f{x). 
Draw  the  ordinates  MA,  NB.  Divide  the  distance  AB  into  n  equal  parts,  and 
construct  upon  these  a  series  of  inscribed  and  circumscribed  rectangles.  The  dif- 
ference between  the  sum  of  the  inscribed  and  the  sum  of  the  circumscribed  rec- 
tangles can  be  seen,  by  simple  inspection,  to  be  =  the  rectangle  ON.  But,  by  in- 
creasing  «,  we  can  make  the  area  of  this  rectangle  approach  o  as  a  limit.  If, 
now,  the  difference  between  the  sum  of  the  inscribed  and  the  sum  of  the  circum- 
scribed rectangles  may  be  made  as  small  as  desired,  it  follows  that  the  difference 
between  the  sum  of  the  inscribed  rectangles  and  the  area  of  the  curve  (which  is  at 
once  greater  than  the  sum  of  the  inscribed  and  less  than  the  sum  of  the  circum- 
scribed rectangles)  may  be  made  to  approach  o  as  a  limit.  Hence  the  area  of  the 
curve  can  be  accurately  found  by  computing  the  limit  to  which  the  sum  of  the  in- 
scribed rectangles  tends,  when  n  becomes  infinite,  and  the  bases  infinitesimal. 
This,  then,  is  clear  warrant  for  our  stating  above  that  y  may  be  treated  as  constant 
over  the  infinitesimal  distance  or  interval  dx. 
D 


50  The  Metric  Methods 

we  obtain  a  more  complicated  formula  for  the  relative  frequency 
of  errors  between  given  fixed  limits.  For  errors,  e.  g.y  between 
the  limits  o  and  +;r  it  runs  : 


P  = 

where  /  is  the  sign  of  summation  or  integration. ^     Similarly, 
the  relative  frequency  of  all  errors  larger  than  this  x  is : 


00 


00 


,— /5  V 


P   =-^  I    e-'^'^dx. 

If  we  put  hx=ty  and  hence  hdx  =  dty  these  last  two  equations 
may  be  written  in  the  form  : 

P'^=~  f  e-^' dt,  3,nd 

00  ^00 

P    =—f  e-^'dt,^ 

Similar  formulae  may  be  written  for  —  x  and  —  oo  .  Since 
the  error  curve  is  symmetrical,  the  probability  that  an  error  lies 
between  the  limits  -\-x  and  —x  is  double  the  probability  that  it 
lies  between  the  limits  o  and  +;ir  or  o  and  —x.     Hence  : 

P     --^  I    e-^dt. 


P       =^-^   C^e-^dt. 


Finally,  the  probability  of  all  errors  whatsoever,  between  the  lim- 
its +  00  and  —  oo  is  : 

+  00^^    f 

The  right-hand  member  of  this  equation  is  given  in  the  text- 
books as  the   *  probability  integral,'  and   is   generally   written 

-^  r  e~^^dt.     Its  value  is  unity.     For,  if  P  be  the  probabil- 
ity  that  an  event  will  happen,  i  —  P  is  the  probability  of  its  fail- 

1  The  f  of  this  and  later  equations  is  merely  a  modification  of  S,  the  initial  let- 
ter of  the  word  '  sum.'  It  denotes  '  limit  of  sum.'  The  actual  sum,  before  passing 
to  the  limit,  is  denoted  by  S  .  Thus  the  sum  of  the  inscribed  rectangles  of  the 
previous  Note  is  written  as  S^A^ ;  the  limit  of  this  sum,  the  area  of  the  curve, 
is  written  f  ydx  or  f  f  {x)  dx. 

^  This  expression  is  tabulated  in  the  mathematical  books  as  the  '  error  function,* 
and  is  denoted  by  Erf  {i).  The  right-hand  member  of  the  equation  immediately 
preceding  is  termed  the 'error  function  complement,' and  is  denoted  by  JEJiT^^  (/). 
See  J.  W.  L.  Glaisher,  Phil.  Mag.,  4  Ser.,  xlir.,  1871,  296,  421. 


§  15-      The  Law  of  Error  51 

ing;  if  the  probability  of  drawing  a  prize  in  a  lottery  be  ^o"o>  ^^ 
probability  of  not  drawing  a  prize  is  |^^  ;  and  P  +  ( i  —P)  = 
2 0^0 0  +  2000  =  ^-  Since  the  probability  integral  sums  up  all  the 
probabilities   of    occurrence,    for    and   against,    we   may   write 

2       /•  00 

7=   I      e—^dt=  I.     This  result  can  be  verified  independently, 

V    TT     J    o 

if  we  evaluate  the  integral  by  the  methods  of  the  calculus.^ 

It  is  not  necessary  that  the  reader  understand  the  above  formulae,  in  the 
sense  that  he  know  how  to  work  with  them,  to  integrate  a  differential  equa- 
tion, etc.;  but  it  is  necessary,  if  the  procedure  of  the  'error  methods'  in 
quantitative  psychology  is  to  be  anything  more  than  a  set  of  blind  rules,  that 
he  understand  in  general  terms  the  reasoning  upon  which  they  are  based. 
That  reasoning,  after  all,  is  not  very  difficult.  The  above  discussion  should 
become  clear  after  two  or  three  readings.  If  it  does  not,  if  there  is  an 
obstinate  hitch  at  some  point  of  the  argument,  the  Instructor  must  be  called 
upon  to  smooth  things  out,  or  to  give  references  for  further  reading. — 

An  illustration  will  show  the  close  accordance  of  theory  with  empirical 
result,  when  the  observer  is  skilled  and  the  number  of  observations  con- 
siderable. The  following  Table  gives  the  distribution  of  errors,  observed 
and  calculated,  in  the  case  of  observations  of  the  differences  of  right  ascen- 
sion of  the  sun  and  two  stars,  a  Aquilae  and  a  Canis  minoris.  The  observa- 
tions were  470  in  number,  and  were  made  by  the  English  astronomer 
J.  Bradley  (1693-1762),  whom  Laplace  called  a  '  model  observer.' 


gnitude  of  errors 

Number  of 

errors : 

in  sec. 

Observed. 

Calculated. 

0.0  —  0.1 

94 

94.8 

0.1  — 0.2 

88. 

88.8 

0.2 — 0.3 

78 

78.3 

0.3  —  0.4 

58 

64.1 

0.4  —  0.5 

SI 

49-5 

0.5  —  0.6 

36 

35-8 

0.6  —  0.7 

26 

24.2- 

0.7  —  0.8 

14 

154 

0.8  —  0.9 

10 

91 

0.9 I.O 

7 

S-o 

Above  1 .0 

8 

5-0 

470        .  470.0 

/-f-    00  


J—C^'^-r^  -J- 

Hence    , —  I         e       dt=i.     The  constant  factor  in    , —  our    original  formula 

was,  indeed,  selected  because  this  factor  is  necessary  to  make  the  total  probability 
unity.     Mathematicians  worked,  at  first,  with  an  indeterminate  constant  Jk, 


52  The  Metric  Methods 

Three  special  points  remain  to  be  noticed. 

( 1 )  We  have  spoken  of  the  error  o  as  the  *  most  probable  * 
error.  Mathematicians  use  the  phrase  *  probable  error,'  in  a  tech- 
nical and  perhaps  somewhat  misleading  sense,  to  denote  that 
magnitude  of  error  which  is  as  often  exceeded  as  not  reached. 
We  may  define  the  probable  error  in  various  ways.  Thus,  if  we 
were  to  arrange  all  the  errors  in  the  order  of  their  magnitude, 
the  probable  error  is  that  which  stands  midway  in  the  series. 
Again,  since  the  probability  integral  is  ==  i,  the  probable  error  is 

that  magnitude  of  x  for  which  the  integral  ~:iz    /    e~^  dt   in 

which  t  =  hx,  has  the  value  o.  5.  Or  again,  it  is  that  magnitude 
of  error  for  which  the  chances  are  even,  one  to  one.  Thus,  if 
5.45  be  the  arithmetical  mean  of  all  determinations  of  the  density 
of  the  earth,  and  the  probable  error  be  .20,  the  meaning  is  that  the 
probability  of  the  actual  density  of  the  earth  falling  between  5.25 
and  5.65  is  one  to  one,  z.  e.,  one  out  of  two,  or  -.  Geomet- 
rically regarded,  the  probable  error  is  the  abscissa  of  that  ordi- 
nate of  the  error  curve  which  divides  each  of  the  symmetrical 
halves  into  two  parts  of  equal  area.     See  Fig.  1 7. 

It  should  be  fully  understood  that  the  phrase  '  probable  error  * 
is  conventional.  The  error  designated  is  not  the  most  probable  : 
that  would  be  the  error  o,  corresponding  to  the  central  ordinate 
of  the  curve.  Nor  is  it,  in  strictness,  probable  at  all ;  since,  as 
we  have  seen,  the  occurrence  of  any  definite,  previously  assigned 
error  is  infinitely  improbable.  These  facts  do  not,  however,  im- 
pair its  practical  value. 

The  probable  error  may  be  very  simply  calculated.  We  find, 
from  the  tables  of  probability  in  the  text-books,  that  /*  =  0.5 
when  hx  =  0.4769,  z.  e.,  when  a  certain  relation  obtains  between 
degree  of  precision  and  magnitude  of  error.     Hence  x,  in  this 

case  PEij  the  probable  error  of  the  single  observation,  is  ^^^, 

or  2.10  PBi  =  ^.  The  measure  of  precision,  ^,  thus  varies  in- 
versely as  PEi. 

(2)  We  must  say  a  word  in  explanation  of  the  value  hx  or  t. 
You  will  presently  be  required  to  use  a  Table,  in  which  various 
numerical  values  are  assigned  to  /  in  the   probability  integral 


2       r*  * 


§  15.     T/ie  Law  of  Error  53 

e~^^dt.      From   these  values,  you  will    be  required   to 


calculate  //,  the  measure  of  precision,  and  the  corresponding  DL» 
It  is  essential,  then,  that  you  understand  what  /  really  is.^ 

The  value  kx  is,  of  course,  the  product  of  the  measure  of  pre- 
cision into  a  determinate  magnitude  of  error.     This  product  is 

identical  with  the  ratio  ^  '-  \\  that  is,  it  is  x  measured  in  terms 

of  ^  .     Now   y  is  called,  technically,  the  *  modulus '  of  the  curve 

of  error :  a  *  modulus '  being,  in  general,  any  constant  coefficient 
or  multiplier  whereby  a  given  series  or  system  of  quantities  may 
be  reduced  to  another  similar  series  or  system.  When,  there- 
fore, you  are  looking  up  a  value  of  t  in  the  Table,  you  are  asking 
how  many  times  (whole  or  fractional)  the  given  error  x  contains 
the  modulus.  If  /  =  i,  the  error  is  equal  to  the  modulus;  if  / 
=  1.5,  the  error  is  half  as  great  again  as  the  modulus ;  if  /  = 
0.5,  the  error  is  half  the  modulus.  In  all  cases,  the  modulus  is 
the  unit  to  which  you  are  referring. 

To  make  the  idea  of  the  modulus  still  more  concrete,  let  us 
call  to  mind  the  fact  that  it  is  =2.1  PEi.  In  looking  up  ^,  then, 
you  are  also  asking  how  many  times  the  given  x  contains  (ap- 
proximately) twice  the  probable  error. 

(3)  Our  last  point  is  of  a  psychological  nature.  It  is  tempt- 
ing to  regard  the  phrase  *  degree  of  probability '  as  equivalent  to 

*  degree  of  expectation '  or  *  amount  of  belief.'  We  give  expres- 
sion, in  daily  life,  to  various  degrees  of  expectation  :  we  say  that 
an  event  is  *  practically  certain,'  *  very  probable,'  *  not  unlikely,' 

*  as  likely  as  not,'  *  hardly  possible,'  *  almost  beyond  belief.'  Do 
these  judgments  correspond  to  certain  numerical  fractions,  rep- 
resenting degrees  of  objective  or  mathematical  probability  }  Is 
the  mathematical  scale  of  probabilities  simply  a  more  finely 
graded  scale  of  subjective  degrees  of  expectation } 

1  The  following  Table  illustrates  the  facts  that  the  integral  (which  we  will  term  /) 
varies  between  the  limits  o  and  i ,  and  that  it  rapidly  approaches  the  latter  limit  as 
/  increases : 

t  I  t  I 

0.75  0.7 1 1  1.50  0.966 

1. 00  .843  1.75  .987 

1.25  .923  2.00  .995 


t 

I 

0.00 

0.000 

.25 

.276 

.50 

.520 

54  The  Metric  Methods 

The  first  thing  that  strikes  us,  when  we  attempt  to  answer 
these  questions,  is  that  the  graduation  of  the  objective  scale  is  in- 
comparably more  accurate  than  that  of  the  subjective.  Intro- 
spection distinguishes,  perhaps,  a  dozen  different  degrees  of 
expectation, — though  it  is  more  than  doubtful  if  the  distances 
between  these  degrees  are  equal,  and  if  the  words  expressing 
them  convey  the  same  meaning  to  different  persons,  or  even  to 
the  same  person  at  different  times.  But  can  we  say  that  the 
expectation   of   an  event  whose   probability  is  -i-  differs  from 

that  of  an  event  whose  probabihty  is  —  .?  Yet  these  are  large 
numbers !  Or  have  we  any  expectation  at  all  that  corresponds 
to  the  probability  ^-^.?  Surely,  we  either  drop  all  thought 
of  the  event,  or  we  have  a  degree  of  expectation  that  is  far  too 
high, — so  high  that  it  cannot  be  distinguished,  say,  from  the  ex- 
pectation of  a  probability  of  — ,  ten  times  as  great.     It  seems 

absurd,  even  if  it  were  theoretically  justifiable,  to  correlate 
two  scales  of  this  kind:  the  one,  of  extreme  and  refined  ac- 
curacy, and  the  other,  as  crude  and  inconstant  as  a  scale  can 
well  be. 

Nevertheless,  it  may  be  objected,  there  is  no  disproof  of  the 
correlation.  It  is  conceivable  that  a  single  degree  of  expectation 
corresponds  to  a  zone  of  degrees  of  mathematical  probability. 
It  is  possible,  too,  that  our  degrees  of  expectation  may  some  day 
be  objectively  measured,  so  that  we  shall  be  able  to  equalise  the 
introspective  distances  between  degree  and  degree,  and  to  assign 
a  numerical  value  (in  terms  of  some  arbitrary  unit)  to  the  degree 
of  expectation  with  which  O  approaches  a  given  series  of  experi- 
ments. We  should  then  have  a  subjective  scale  of  coarse  adjust- 
ment, and  an  objective  scale  of  fine  adjustment.  Why  might 
not  the  two  be  correlated  .? 

The  reply  is  twofold.  In  the  first  place,  there  is  no  practical 
ground  for  beheving  that  the  correlation  can  ever  be  made  out. 
Expectation  is  largely  a  matter  of  feeling,  of  temperament :  the 
sanguine  mind  will  confidently  expect,  positively  anticipate,  an 
event  which,  to  the  phlegmatic  mind,  appears  exceedingly  im- 
probable. It  is  difficult  to  see  how  this  typical  difference  of 
mental  constitution  could  be  overcome.     In  the  second  place. 


§  1 6.    TJie  Method  of  Limits  55 

there  is  no  theoretical  warrant  for  the  correspondence.  On  the 
contrary,  the  two  scales  are  incommensurable.  For  it  is  of  the 
essence  of  expectation  (as  it  is  of  the  kindred  states  of  attention, 
practice  and  fatigue)  that  it  ranges  between  fixed  limits.  There 
is  a  maximal  degree  of  expectation.  Just  as  we  find  a  terminal 
stimulus,  in  the  psychology  of  sensation,  such  that  further  ob- 
jective change  has  no  subjective  concomitant,  so  do  we  find  a 
certain  terminal  degree  of  attention,  expectation,  practice,  beyond 
which  our  concentration,  certainty,  facility,  cannot  be  enhanced. 
We  '  feel  sure,'  our  expectation  becomes  conviction,  long  before 
we  have  the  objective  right  to  be  anything  more  than  very  mod- 
erately expectant.  If,  then,  we  are  to  correlate  probability  with 
belief,  objective  chance  with  subjective  expectation,  we  can  do  so 
only  in  the  sense  that  we  correlate  it  with  a  *  rational  expectation,' 
with  *  belief  as  it  ought  to  be.'  But  this  sort  of  expectation  and 
belief  is  not  the  sort  that  psychology  is  acquainted  with ;  the 
attempt  at  correlation  leads,  inevitably,  to  an  ideal  reconstruction 
of  mind ;  and  therefore,  from  the  psychological  point  of  view, 
stands  self -condemned. 

§  1 6.  The  Method  of  Limits  (Method  of  Just  Noticeable  Differ- 
ences; Method  of  Least  Differences ;  Method  of  Mimimal  Changes). — 
Let  us  suppose  that  we  are  to  determine  the  j.  n.  d.  of  bright- 
ness ;  that  we  are  to  measure  the  DL  at  various  points  upon  the 
brightness  scale.  All  greys,  as  we  know.  He  upon  a  single  straight 
line  between  the  limits  B  and  W  {cf.  the  colour  pyramid  :  vol  i., 
3).  We  may,  then,  select  any  convenient  point  upon  this  Hne  as 
the  starting-point  of  our  investigation. 

The  arrangement  of  the  experiment  is  simple.  We  take  an 
ordinary  colour  mixer,  and  mount  upon  it  two  pairs,  large  and 
small,  of  black  and  white  discs.  One  disc  of  each  pair  must  be 
accurately  graduated  on  the  back  in  1°  units ;  the  gradua- 
tion may  extend,  perhaps,  over  45°.  We  are  thus  able  to 
change  the  relation  of  the  black  and  white  sectors,  over 
a  range  of  45°,  by  units  or  half  units,  1°  or  0.5°  Since 
we  are  not  bound  down  to  any  particular  point  of  departure,  we 
may  give  both  of  the  compound  discs  an  initial  value  of  180**  W 
and   1 80°  B.     If  the  four  discs  have  been  cut  from  the  same 


56  The  Metric  Methods 

two  sheets  of  card  or  paper,  and  if  the  edges  of  the  smaller  pair 
are  accurately  trimmed,  we  shall  see  on  rotation  a  perfectly  uni- 
form lightish  grey.     How  are  we  to  determine  the  j.  n.  d.  ? 

The  answer  appears  as  simple  as  the  arrangement  of  appara- 
tus. We  have,  it  would  seem,  only  to  lighten  the  one  or  other 
of  the  greys,  adding  in  white,  i°  at  a  time,  in  successive  observa- 
tions, until  O  says  *  Lighter  ! '  Then  we  set  down  in  our  record 
the  composition  of  the  new  grey,  which  is  j.  n.  lighter  than  the 
first  grey  of  1 80°  W  and  1 80°  By — and  the  problem  is  solved. 

The  quickest  way  is,  however,  not  always  the  safest.  And  a 
little  reflection  upon  the  results  of  the  experiments  already 
performed  will  suggest  that  we  have  here  gone  to  work  incon- 
siderately. We  have  not  tried  to  get  a  clear  idea,  before  starting 
on  the  experiment,  of  what  the  DL  really  is.  Is  it,  e.  g.y  an  S- 
distance  that  has  a  definite  mental  equivalent ;  that  we  can  carry 
in  our  heads,  as  a  pattern  to  go  by ;  that  we  can  remember,  from 
experiment  to  experiment,  so  that  we  shall  always  obtain  the 
same  result,  always  stop  the  addition  of  W  at  the  same  point  i^ 
Or  is  it  rather  an  ideal,  calculated  value  ;  a  distance  that  has  no' 
distinct  mental  representation,  that  we  are  not  called  upon  to 
estimate  or  to  remember,  but  that  must  be  established,  at  the 
conclusion  of  each  experiment,  by  mathematical  treatment  of 
the  results .?  A  very  important  point  !  And  again  :  we  have 
not  asked  ourselves,  before  starting,  whether  the  arrangement  of 
our  experiment  closes  the  door  upon  all  the  many  errors  to 
which,  as  preliminary  experiments  have  taught  us,  psychological 
observation  is  liable.  For  all  that  we  know,  it  may  actually 
favour  certain  sorts  of  error.  Another  important  point ;  and 
one  that  must,  surely,  be  considered  before  experimental  work 
is  begun  ! 

Let  r  represent  the  standard  or  normal  stimulus  in  our  experi- 
ment, /.  e.y  a  compound  disc  of  1 80°  B  and  1 80°  W.  This  is  to 
remain  constant  throughout.  Similarly,  let  r^  represent  the  vari- 
able stimulus  or  stimulus  of  comparison,  i.  e.y  a  compound  disc 
whose  sectors  are  to  be  changed,  from  observation  to  observation, 
until  the  resulting  grey  is  j.  n.  lighter  than  the  standard  grey. 
The  first  thing  to  notice  is  that  r  and  r^  have  different  positions 


§  1 6.      The  Method  of  Limits  57 

in  space.  If  we  had  performed  the  rough  experiment  indicated 
above,  we  should  probably  have  put  r  on  the  inside  and  r^  on  the 
outside,  since  it  is  easier  to  manipulate  the  large  than  the  small 
discs.  But  it  makes  a  difference  to  our  j.  n.  d.,  whether  r^  is  on 
the  outside  or  on  the  inside.  Why  it  should  make  a  difference 
is  not  easy  to  say.  If  the  greys  were  set  up  on  different  mixers, 
we  might  suspect  differences  of  physical  illumination,  or,  perhaps, 
differences  in  the  physiological  condition  of  the  retina.  Here,, 
however,  the  two  greys  are  juxtaposed,  and  can  be  viewed  with 
strict  simultaneity.  It  may  be  that  the  reason  for  the  influence 
of  spatial  position  lies  in  the  changed  attitude  of  judgment  itself,, 
in  a  different  set  of  the  attention  over  against  differently  ar- 
ranged R.  However  this  may  be, — and  we  are  compelled  to 
leave  the  question  unsettled, — any  single  series  of  observations 
will,  in  actual  fact,  be  vitiated  by  a  *  constant  error,  the-  space  er- 
ror.  Fortunately,  the  space  error  can,  in  this  case,  be  eliminated. 
If  we  have  determined  a  DL  with  r  on  the  inside  and  r^  on  the 
outside,  we  have  merely  to  make  a  similar  determination  with  r 
on  the  outside  and  Vy  on  the  inside,  and  to  average  the  two  results. 
The  space  errors  are  opposed,  in  the  two  cases,  and  will  cancel 
out  when  we  average. 

The  second  point  to  notice  is  that,  when  O  is  called  upon  to 
make  a  series  of  observations  leading  to  a  single  result, — /.  e.y 
when  the  experiment  is  progressive,  and  requires  a  certain  amount 
of  time  for  its  performance, — his  consciousness  is  apt  to  take  on 
a  definite  trend  or  disposition,  which  may  affect  the  outcome. 
Thus,  the  longer  O  works,  the  more  practised  will  he  become ; 
and  the  more  practised  he  becomes,  the  smaller  (other  things 
equal)  will  his  DL  be,  until  the  limit  of  maximal  practice  is 
reached.  On  the  other  hand,  the  longer  he  works,  the  more 
fatigued  will  he  become  ;  and  the  more  fatigued  he  becomes,  the 
larger  (other  things  equal)  will  his  DL  be.  Practice  and  fatigue 
thus  work  against  each  other.  Again,  if  E  begins  gradually  to 
lighten  ri,  O  may  be  expecting^  from  observation  to  observation, 
that  the  lightening  will  become  apparent  in  5 ;  and  the  farther 
the  series  has  gone,  the  greater  will  this  expectation  be.  But  ex- 
pectation means  anticipatory  attention,  an  outrunning  of  fact  by 
judgment.     Hence  0  will  tend  to  make  the  difference  come  too 


58  The  Metric  Methods 

soon  ;  he  will  declare  that  ?i  is  lighter  than  r  when,  apart  from 
the  influence  of  expectation,  he  would  still  have  taken  the  two 
greys  to  be  alike.  Contrariwise,  if  the  series  has  already  pro- 
gressed a  certain  distance,  and  O  has  been  wholly  unable  to  find 
a  difference  between  r  and  ri,  he  may  be  led  to  think  that  the 
coming  change  is  still  some  few  steps  away ;  so  far  from  expect- 
ing a  difference,  he  will  be  expecting  a  likeness  of  the  two  R.  His 
expectation  is  still  an  anticipatory  attention  ;  but,  in  this  form,  it 
tends  to  make  the  observed  difference  come  too  late.  Finally, 
and  quite  apart  from  expectation,  the  farther  the  series  has  gone, 
the  more  accustomed  has  O  become  to  saying  *  like,'  and  the 
more  likely  is  he  to  continue  to  say  the  same  thing.  A  habit  of 
judgment  has  been  set  up ;  attention  lags  behind  the  changes  of 
ri.  Hence  O  will  again  tend,  other  things  equal,  to  make  the 
difference  come  too  late ;  he  will  find  two  greys  alike  that  he 
would  otherwise  see  to  be  different.  Habituation  is  thus  opposed 
to  the  first  form  of  expectation,  while  it  works  in  the  same  di- 
rection as  the  second. 

All  these  sources  of  error — practice,  fatigue,  expectation  in  its 
two  forms,  habituation — are  variable^  in  the  sense  that  the  state 
or  degree  of  each  one  of  them  varies  with  the  progress  of  the 
series.  What  their  combined  effect  would  be,  in  a  particular 
case,  we  cannot  possibly  say.  But  at  any  rate  we  cannot  count 
upon  their  offsetting  one  another.  They  are  as  dangerous  and 
as  inevitable  as  is  the  *  constant '  space  error  ;  and  we  must  take 
account  of  them,  and  seek  to  remove  or  to  minimise  them,  if  our 
result  is  to  be  reliable. 

Lastly,  the  experiment  is  subject  to  errors  which,  like  the  five 
just  mentioned,  vary  from  point  to  point  of  the  series,  but  which, 
unlike  these  five,  do  not  vary  regularly  or  continuously  with  the 
course  of  the  work.  We  may  call  them  accidental  errors.  They 
arise,  e.  g.,  from  fluctuation  or  slipping  of  the  attention  under 
subjective  influences,  from  variations  in  (9's  general  mood  or 
state  of  health,  from  lapse  of  attention  under  outside  distrac- 
tions, from  errors  of  manipulation  on  E'^  part,  from  all  sorts  of 
physiological  causes.  We  cannot  treat  them  in  the  same  way 
that  we  treat  practice  and  the  rest,  but  we  must  in  some  way 
take  account  of  them. 


§  1 6.      The  Method  of  Limits  59 

It  follows  from  this  account  that  the  value  which  we  find  for 
the  DL  in  a  single  series  of  observations  is  practically  worth- 
less. It  is  vitiated  by  a  constant  error ;  it  is  affected,  on  this 
side  or  that,  by  one  or  more  of  the  variable  errors ;  it  is  the 
sport  of  the  accidental  errors.  The  true  DL  is,  then,  an  ideal 
value,  never  directly  determinable  ;  it  is  measured  by  that  incre- 
ment of  R  which,  after  complete  elimination  of  all  errors,  con- 
stant and  variable  alike,  would  make  a  given  ri  j.  n.  d.  from  r  in 
the  judgment  of  an  accurate  O.  But  we  can  never,  in  a  single 
series,  eliminate  the  constant  error ;  we  can  never,  except  by  the 
merest  chance, — and  then  we  should  not  know  that  we  had  done 
it ! — strike  an  exact  balance  among  what  we  have  called  the  va- 
riable errors  ;  and  we  can  never  wholly  rid  ourselves  of  the  acci- 
dental errors.  The  DL  must  be  calculated  from  the  results  of 
a  number  of  series,  carefully  planned  and  disposed ;  and,  even 
so,  its  value  is  only  approximative,  valid  under  the  total  condi- 
tions of  the  experiment. 

'  Errors  of  observation '  have  been  treated  by  mathematicians  in  what  is 
called  the  '  theory  of  probabilities.'  From  the  mathematical  point  of  view, 
we  may  measure  the  true  DL  by  that  increment  of  R  which  (after  elimina- 
tion of  constant  errors)  enables  O  correctly  to  distinguish  between  r  and  r^ 
in  50  %  of  a  long  series  of  observations,  while  in  the  remaining  50  % 
his  judgment  of  the  relation  of  r  and  r-i  is  either  uncertain  or  posi- 
tively wrong.  This  definition  of  the  DL  is  not  different  from  that  given 
above :  only,  the  former  definition  regarded  the  variable  errors  as  ruled  out, 
while  the  mathematical  definition  leaves  these  errors  in,  and  treats  them  by 
the  laws  of  probability. 

Suppose,  now,  that  we  keep  to  our  original  idea  of  changing 
ri  by  small  amounts,  in  order  to  make  it  j.  n.  lighter  than  r,  but 
that  we  work  out  this  idea  methodically,  with  a  view  to  eliminat- 
ing or  minimising  the  errors  of  observation.  How  will  our 
method  take  shape .? 

In  the  first  place,  we  must  adapt  the  method  to  our  revised 
idea  of  the  DL.  We  have  conceived  the  method  to  begin  with 
an  ri  that  is  =  and  W^r;  and  we  have  thought  of  this  ri  as  grad- 
ually lightened,  up  to  the  point  at  which  the  first  judgment  of 
<  lighter '  is  given.  Should  w-e,  by  this  procedure,  determine  the 
true  DL  ?     Surely  not.     The  DL  stands  on  the  border-line  be- 

1  This  symbol  means  '  subjectively  equal  to  ',  as  =  means  '  objectively  equal  to  \ 


6o  The  Metric  Methods 

tween  noticeable  and  unnoticeable  differences,  and  we  have  de- 
termined only  a  least  noticeable  difference.  We  must,  accord- 
ingly, take  two  determinations,  from  opposite  directions.  Starting 
•out  from  an  r^  that  is  noticeably  lighter  than  r,  we  gradually 
"darken  it,  until  its  difference  from  r  ceases  to  be  noticeable  (until 
O  says  '  same  ',  or  *  doubtful ',  or  perhaps  '  darker  ').  Then, 
:starting  out  from  an  7\  that  is  1 1 1  r,  we  gradually  lighten  it,  until 
^  says  *  lighter '.  Finally,  we  average  the  results  of  the  two 
series, — the  r^  —  r  which  gave  ;i  as  j.  not  -n.  lighter  than  r  in  the 
ifirst,  and  the  rj  —  r  which  gives  ^i  as  j.  n.  lighter  than  r  in  the 
second, — to  obtain  a  true  DL. 

Secondly,  we  must  take  measures  for  the  avoidance  of  errors. 
The  elimination  of  the  constant  space  error  has  already  been 
discussed.  Whatever  we  do  with  r  inside  and  r^  outside,  we  must 
repeat  with  r  outside  and  r^  inside.  The  whole  experiment  must 
be  performed  twice,  with  reversal  of  the  space  relations  of  r  and 
Txy  and  the  average  of  the  two  results  taken.  The  space  error 
just  doubles  our  work.  Variable  errors  are  reduced  to  a  mini- 
mum, partly  by  directions  given  to  (9,  partly  by  a  judicious  ar- 
rangement of  the  series.  Accidental  errors  are  compensated  by 
frequent  repetition  of  the  paired  series.  The  results  of  all  series 
are  averaged  to  a  final  DL,  and  this,  recorded  along  with  a  meas- 
ure of  its  variability,  affords  the  solution  of  our  problem. 

It  is  clear  that  we  are  here  applying  the  method  of  Exp.  I.,  so  modified  as 
to  furnish  a  DL  in  place  of  an  RL.  It  is  unnecessary  to  repeat  the  de- 
scription of  the  method  in  full :  the  following  brief  indications  will  suffice. 
See  above,  pp.  4  £f. 

Directio7is  to  E. 

(i)  Preliminary  experiments. — The  approximate  position  of  O's  DL  must 
always  be  determined  by  preliminary  experiments,  since  it  is  advisable 
to  begin  the  experiment  with  a  \  series. 

(2)  Size  of  steps. — The  steps  must  in  every  case  be  small.  They  should 
be  kept  constant  within  a  (paired)  series.  On  the  other  hand,  they  may 
vary,  as  between  one  (paired)  series  and  another.  Thus,  in  the  illustration 
:taken,  they  might  be  made  1°  in  the  one  half  of  the  series,  and  0.5°  in  the 
•other.  If  we  were  working  not  with  B  and  W^  but  with  two  not  very  dif- 
ferent greys,  the  range  of  variation  would  be  much  wider. 

(3)  Starting-point  of  the  series. — We  have  assumed  that  the  ^  series  starts 
from  the  objective  and  subjective  equality  of  r  and  r^.     Now  («),  owing  to 


§  1 6.      TJie  Method  of  Limits  6 1 

constant  errors,  the  two  do  not  always  coincide.  If  the  two  compound  discs 
of  1 80°^  and  180°^-^  were  set  up,  e.  g.^  on  separate  mixers,  it  might  very 
well  happen  that  they  did  not  appear  equal  to  O.  In  such  a  case,  we  must 
give  up  the  objective  and  keep  to  a  subjective  equality.  But  again,  {b)  the 
DL  may  be  so  variable  that  it  is  impossible  to  find  an  n  that  always  ap- 
pears equal  to  the  standard  r.  In  such  a  case,  we  must  give  up  the  idea  of 
equality  altogether,  and  start  out  from  an  ri  that  either  appears  uniformly 
darker  than  r,  or  appears  (according  to  circumstances)  now  darker  than 
r,  now  equal  to  it.  {c)  In  any  event,  whether  we  keep  within  the  limits  of 
subjective  equality,  or  trespass  upon  the  '  darker '  ri,  it  is  advisable  to  vary 
the  point  of  departure  of  the  \  series.  The  starting-point  of  the  |  series 
(an   rx  that   is  always   judged  lighter  than   r)  should  be  similarly  varied. 

(4)  Order  of  series. — The  experiment,  on  the  analogy  of  Exp.  I.,  will 
consist  of  40  series:  10  paired  series  in  each  of  the  two  space  positions. 
They  must  be  so  arranged  that  the  effects  of  practice  are  distributed,  as 
■evenly  as  possible,  over  the  whole  experiment. 

(5)  Length  of  successive  series. — Series  of  three  lengths  are  to  be  used: 
3  long,  4  moderate,  3  short  in  every  set  of  10.  Their  order  is  determined 
by  chance. 

Directions  to  O. 

(i)  Direction  of  judgment. — Judgment  is  always  given  in  terms  of  r\. 
Thus,  if  O  says  '  lighter,'  he  means  that  rx  appears  lighter  than  rj  if  he 
says  '  doubtful,'  he  means  that  he  cannot  tell  whether  r\  is  lighter  than, 
equal  to,  or  darker  than  ry  and  so  on. 

(2)  Variable  errors. — It  is  essential  that  O  be  on  his  guard  against  the 
variable  errors  of  expectation  and  habituation.  He  must  judge  every  pair 
of  /?,  as  it  comes,  with  maximal  attention  and,  so  far  as  possible,  without 
any  reference  to  previous  judgments.  He  must  not  try  consciously  to  cor- 
rect the  errors  ;  he  must  not,  e.g..,  suspect  himself  of  an  expectation  error,  and 
try  to  overcome  it  by  an  effort  of  will :  he  must  simply  look  attentively  at  the 
greys,  as  they  are  presented,  and  make  up  his  mind,  without  prejudice,  as 
to  the  relation  of  r^  to  r. 

(3)  The  error  of  bias. — If  O  allows  his  attention  to  relax,  and  if  E  does 
not  sufficiently  vary  the  series,  we  find  a  new  source  of  error,  introduced 
into  the  results  by  the  form  and  course  of  the  method  itself,  which  we  may 
term  the  error  of  bias.  The  indefinite  expectation,  which  the  method 
presupposes  and  takes  account  of,  gives  place  to  a  definite  anticipation  of 
change  at  some  particular  step  in  the  series  :  O  looks  for  a  turn  of  judgment 
at  the  fourth,  fifth,  etc.,  observation.  Should  E  or  O  suspect  this  bias,  the 
haphazard  arrangement  of  the  series-lengths  must  be  given  up  ;  E  must,  for 
a  little  while,  use  only  the  longest  and  shortest  series.  If  the  bias  has  gone 
very  far,  he  may  introduce,  as  circumstances  suggest,  a  very  long  or  a  very 
short  series.  Such  a  series,  it  must  be  remembered,  has  a  disciplinary  value 
only :  it  is  foreign  to  the  method ;  it   interrupts  the  regular  course  of  the 


62  The  Metric  Methods 

experiment ;  its  results  cannot  be  counted  in  with  the  rest.  The  prevention 
of  the  error  of  bias  is  indefinitely  better  and  easier  than  its  cure.  And,  since 
E  already  has  his  directions  with  regard  to  variation  of  the  length  of  the 
series,  its  prevention  lies  in  6>'s  hands. — Cf.  p.  4  above. 

The  \  series  furnishes  an  ri  which  marks  the  point  at  which 
(9*s  judgment  changes  from  *  lighter '  to  '  same',  *  doubtful '  or 
*  darker.'  This  value  of  r-^  is  recorded  as  r^  {d=  descending).  The 
^  series  furnishes  an  r^  which  marks  the  point  at  which  6>'s  judg- 
ment changes  from  *  same  '  or  *  doubtful '  to  *  lighter.'  This  value 
of  ri  is  recorded  as  r„  (^=ascending).  The  difference  r^ — r  is 
recorded  as  Ar^;  the  difference  r„ — r  as  Ar„.  The  average  of 
A^d  and  A^a  is  written  A^.  The  final  A^  is  the  average  of  all 
the  separate  A^^  and  A^a  values.  If,  as  was  suggested  above, 
we  take  10  paired  series,  in  each  of  the  two  positions  of  space,  the 
final  A  ^  will  therefore  be  the  average  of  40  determinations. 

We  have  now,  if  we  have  worked  aright,  a  value  of  the  DL 
(this  final  A?')  which  is  unaffected  by  any  constant  error,  and  is 
as  free  from  variable  and  accidental  errors  as,  in  the  time  at  our 
disposal,  we  can  make  it.  The  tale  of  errors  is  long ;  and  the 
reader  who  has  followed  the  preceding  discussion  may,  perhaps, 
be  inclined  to  distrust  a  method  that  has  to  move  so  warily  among 
so  many  pitfalls.  But  the  sources  of  error  are  there :  we  must 
either  avoid  them  or  fall  into  them  :  and  that  is,  surely,  the  best 
rule  of  work  which  takes  account  most  fully  and  explicitly  of  the 
dangers  that  beset  the  path  of  the  experimenter.  It  is,  in  reality, 
an  advantage  of  the  method  of  limits  that  the  errors  involved 
are  obvious,  can  be  separately  traced  and  definitely  named  :  for 
that  means  that  they  can,  with  some  little  effort,  be  eliminated. 
And  in  spite  of  the  number  of  errors,  the  course  of  the  method 
in  practice  is  comparatively  simple.  All  that  E  has  to  do  is  to 
grade  his  R  according  to  rule  ;  all  that  O  has  to  do  is  to  compare, 
with  maximal  attention,  every  pair  of  R  presented. 

In  conclusion,  the  information  which  the  method  gives  may  be 
summed  up  under  the  headings  :  magnitude,  course  and  preci- 
sion of  the  DLf  and  magnitude  and  direction  of  the  constant 
error. 

(i)  Magnitude  of  the  DL. — The  absolute  magnitude  of  the 


§    1 6.      The  Method  of  Limits  63 

DL  is  shown  by  the  value  l^r ;  its  relative  magnitude,  i.  c,  its 
magnitude  relatively  to  the  magnitude  of  the  standard  stimulus, 

Ar 
by  the  value  — . 

(2)  Course  of  the  DL. — On  this  point,  we  can  say  nothing. 
We  have,  by  supposition,  determined  only  one  DL  at  one  point 
upon  the  i^-scale.  We  may,  of  course,  compare  our  result 
with  the  published  results  of  investigators  who  have  worked  un- 
der similar  conditions.  Or,  better  still,  we  may  compare  it  with  the 
results  of  other  students  who  have  worked,  at  different  points  on 
the  7?-scale,  in  the  same  laboratory  and  with  the  same  materials. 
It  is  well,  however,  if  time  permits,  to  take  at  least  one  other 
DL^  for  purposes  of  comparison. 

In  a  serious  investigation,  the  determination  of  r^and  Va  would  be  made, 
not  at  one  point  upon  the  7?-scale  (the  grey, e.g.,  of  180°^  and  iSo°lV),  but 
at  several  points  between  its  extreme  limits.  Thus,  in  one  of  the  most  com- 
plete researches  that  we  have,  the  Z>L  of  brightness  was  determined  for  18 
intensities  of  light  between  the  limits  0.5  and  200000:  the  light  unit,  i, 
corresponding  roughly  to  the  illumination  of  a  dull-finish  white  paper  by  a 
stearin  candle  75  cm.  distant,  as  viewed  through  a  screen  opening  of  i  mm^. 
These  18  determinations  allow  us  to  plat  a  curve  of  the  course  of  the  DL 
from  dark  to  light. — Method  work  on  such  a  scale  is,  naturally,  out  of  the 
question  in  a  laboratory  course. 

If  we  decide  to  make  one  other  determination,  we  shall  do  best  to  work 
with  the  same  r,  but  to  vary  ri  on  the  lower  (darker)  side  of  it.  If  we  term 
the  A^  already  determined  the  upper  DL  of  the  r  in  question,  the  new 
Ar  will  be  the  lower  DL.  To  obtain  it,  we  set  out  from  an  r\  that  is  notice- 
ably darker  than  r,  and  gradually  lighten  n  until  O  says  '  same,'  '  doubtful,' 
or  '  lighter.'  The  value  of  the  ?„  is  recorded.  Similarly,  we  set  out  from  an 
ri  that  is  subjectively  equal  to  (or,  at  any  rate,  not  darker  than)  ri,  and 
gradually  darken  r^  until  O  says  '  darker.'  The  value  of  the  the  r^  is  re- 
corded.    Finally,  the  values  of  r —  r«  and  r —  ra  are  averaged  to  a  Ar. 

If,  now,  the  absolute  DL  is  constant,  we  must  find,  for  all  the 
determinations  made  in  the  laboratory  at  different  points  of  the 
i?-scale,  that  A^=<^,  where  ^  is  a  constant  value.  If,  on  the 
other  hand,  the  relative  DL  is  constant,  we  must  find,  through- 
out, that  —  =  k,  where  >^  is  a  constant.  Or  again  :  if  we  average 
r^  and  r„  to  r^  {m=vi\Qdin),  we  must  find  that  ^  =  Cy  where  C  is 


64  The  Metric  Methods 

a  constant.     The  value  —  is  sometimes  termed  a  quotient  limen 

or  QL. — We  thus  have,  in  all,  the  test-values  Ar,  —  ,  *^. 

r       r 
If  we  have  determined  the  lower  as  well  as  the  upper  DL^  we  have  six 
test-values,  as  follows  {u  z=  upper,  /  =  lower) : 

the  absolute  BL Ar„,  Ar^ ; 

the  relative  DL     .     .     ,     .     ,     ^,    — ^!i_; 

r     '    r  —  An 

and  the  gZ !^,    ~. 

(3)  Precision  of  the  Observations. — Psychological  practice  has 
usually  been  content  to  take  the  J/ Fas  the  measure  of  precision 
for  the  results  of  the  method  of  limits.  It  is,  for  some  reasons, 
better  to  give  the  probable  error  or  PE^  a  variation  from  the 
mean  of  such  magnitude  that  any  given  variation  is  as  likely  to 
exceed  it  as  to  fall  below  it.     The  PE  of  'a  single  observation 

may  be  calculated  from  the  formula  PE.^=o.6j^^  4  / — ^,  where  s 

is  the  sign  of  summation,  n  is  the  number  of  observations,  and  v 
stands  for  the  differences  A — ^,  A — b^  etc.,  of  the  formula  on 

p.    8.       The    quantity  i/-^-    is    termed    the    error  of    mean 

square;  it  is  the  error  whose  square  is  the  mean  of  the  squares  of 
all  the  errors.!  In  words,  then,  PE^  =  two-thirds  of  the  error 
of  mean  square.  The  PE  of  the  mean  (of  Ar)  may  be  calcu- 
lated from  the  formula  7^^^=0.6745  4/ — f^^    .     Simpler  but 

y    n  {fi — I) 

somewhat  less  accurate  formulae  are  : 

P£  ^^    0.8453  ^^. 

^  «  (n—i) 

PJ7  0.8453   ^^ 

, • 

n  ^     [n—i) 

1  This  is  the  general  definition  of  the  '  error  of  mean  square.'    If  we  were  to  follow 

it  strictly,  we  should  have,  not  4  / ,  but  4  /    ^"!.    It  is,  however,  shown  in  mathe- 

y  n  —  I  y     n 

matical  books  that,  where  the  number  of  observations  is  small,  we  insure  greater 
accuracy  of  result  by  writing  n  —  i  for  «  in  the  denominator  of  the  fraction. 
Hence  we  are  not  really  running  counter  to  our  definition  ;  we  are  simply  making 
the  error  of  mean  square  the  error  whose  square  is  the  corrected  mean  of  the 
squares  of  all  the  errors.     See,  e.g.^  M.  Merriman,  Least  Squares,  1900,  71. 


§    1 6.    The  Method  of  Limits  65 

It  follows  that,    approximately,  PE^  =  Q.Z^  MVy  and    PEm^ 
°'  ^    MV.     These  latter   equations   are   sufficiently   accurate 


4 


Tt 1 


for  our  purposes. 

(4)  Magnitude  of  the  Constant  Error. — Let  us  term  the  DL 
obtained  with  r  to  the  right  and  r^  to  the  left  A^i,  and  the  DL 
obtained  with  r  to  the  left  and  n  to  the  right  A^n.     Then 

2 

and  qy  the  space  error,  =  — ^ ^'. 

If  we  have  determined  A  r^  as  well  as  A  r„,  we  have 

^^___  Ar..,4-Ar„„^  ^^^  Ar„+Ar,„, 

2  2 


whence 


Ar«,— Ar«„  Ar;„— Ar^, 

^  =: ,         ^= . 


Or  again :  if  we  regard  as  negligible  the  small  difference  (due  to  Weber's 
Law)  between  A  r„  and  A  r^,  we  may  write 


whence 


^  r,  = ,        ^  r,i  :rr j 


Ar^i—  An,                Arz„— Ar„ 
^=  ^  '      ^-  2 


In  this  matter  of  the  constant  error,  the  advantage  of  the  additional  deter- 
mination of  A  ri  is  obvious. 

(5)  Direction  of  the  Constant  Error. — The  space  error  is  equiv- 
alent, for  practical  purposes,  to  a  certain  algebraical  increment 
of  the  difference  D  existing  between  r  and  r^.  In  the  one  ar- 
rangement D  is  increased,  in  the  other  diminished,  by  a  small 
constant  amount.  Fechner  calls  the  space  Qxror  positive  when 
its  effect  is  to  make  the  left-hand  R  appear  greater  than  the 
right ;  negative^  when  its  effect  is  to  make  the  left-hand  R  ap- 
pear less  than  the  right.  The  distinction  is  merely  conventional, 
but  has  remained  current  since  Fechner's  time.  It  is  evident 
that  the  signs  of  the  formulae  just  given  presupppose  a  negative 
space  error :  if  the  error  is  positive,  the  signs  will  be  reversed. 

E 


66  The  Metric  Methods 

EXPERIMENT  XIII 

Determination  of  the  DL  for  Brightness.  —  This  experiment 
may  be  performed  in  various  ways. 

(i)  Materials. — Two  colour  mixers.  Four  large  discs,  two 
black  and  two  white,  all  sUt  along  one  radius,  and  the  latter 
graduated  at  the  back.  Grey  screen,  of  approximately  the  same 
brightness  as  the  compound  discs  employed.  Kirschmann  pho- 
tometer.    Window  screens  of  white  muslin.     Metal  protractor. 

Disposition  of  Apparatus. — The  two  mixers  are  set  up,  side 
by  side,  before  the  grey  screen.  Discs  and  screen  should  be 
directly  illuminated  by  a  high  window,  which  does  not  admit  sun- 
light during  the  hours  of  experimentation  ;  O  sits  with  his  back 
to  the  window.  The  general  illumination  of  the  room  should  be 
kept  so  far  as  possible  constant,  from  day  to  day,  by  the  use  of 
muslin  screens  set  in  the  various  windows. 

(2)  Materials. — Colour  mixer.  Two  large  (black  and 
white)  and  two  small  (black  and  white)  discs,  all  sHt  along  one 
radius,  and  the  white  graduated  at  the  back.  Grey  screen, 
Kirschmann  photometer,  muslin  screens,  protractor,  as  before. 
Black  observation  tube.     Eye-shade. 

Disposition  of  Apparatus. — The  four  discs  are  mounted  on 
the  mixer,  which  stands  as  before  in  front  of  the  screen.  Ob- 
servations may  be  taken  binocularly,  with  the  free  eyes,  or  mo- 
nocularly,  through  the  observation  tube.  This  is  of  black  card- 
board, widening  at  the  near  end  to  fit  the  eye ;  the  reflection  of 
light  from  its  sides  must  be  carefully  avoided.  The  unused  eye 
is  shaded. 

(3)  (4)  Materials  as  in  (i)  and  (2),  except  that  the  black  and 
white  are  replaced  by  two  Hering  greys.  The  advantage  of  us- 
ing greys  is  that  the  serial  steps  are  shorter,  while  the  actual 
changes  of  setting  are  larger,  than  they  are  with  black  and  white, 
so  that  the  DL  can  be  determined  with  greater  accuracy. 

General  Directions. — Readings  should  be  taken,  at  the  end 
of  every  series,  both  from  the  scale  at  the  back  of  the  graduated 
disc,  and  also  from  the  protractor  scale.  The  latter  gives  the 
more  accurate  value ;  and  a  discrepancy  between  the  two  results 
may  serve  to  call  ^'s  attention  to  some  error  of  manipulation 
during  the  course  of  a  series. 


§    1 6.    The  Method  of  Limits  67 

The  photometric  value  of  the  brightnesses  employed  should  be 
determined  by  the  Kirschmann  photometer  (set  up  in  the  same 
position  as  the  mixers)  both  at  the  beginning  and  at  the  end  of 
the  experiment.  Even  if  the  discs  are  carefully  kept  in  a  dark 
drawer  in  the  intervals  between  the  laboratory  sessions,  there  is 
danger  of  fading.  The  two  sets  of  photometric  values  are  aver- 
aged for  the  final  determination  of  the  DL. 

The  resuhs  are,  of  course,  entered  fully  in  the  note-book  in  terms  of  de- 
grees. At  the  end  of  the  experiment,  however,  the  results  of  the  various 
series  must  be  translated  into  photometric  values.  Suppose,  e.g.^  that  a 
black  and  a  white  are  used,  whose  brightness-ratio  is  given  by  the  photome- 
ter as  I  :  40.  Then  the'  value '  of  a  disc  of  180°  B  and  a  180°  ^  is  180  X 
I -j- 1 80X40  or  7380  'photometric  units.'  If  the  amount  of  the  average 
DL  were  5°,  the  value  of  the  corresponding  disc  would  be  1 75  X  i  +  1 85  X  40j 
or  7575  units.     The  relative  DL  would  then  be  expressed  by  the  fraction 

195  I 

^^    or 


7380        38 

Preliminary  experiments  are  to  be  made  before  the  experiment 
proper  is  begun  (p.  60  above).  But  more  than  this  :  E  should 
always  give  O  a  little  practice  at  the  beginning  of  each  session, 
before  entering  on  the  series  laid  out  for  the  day  ;  he  should  on 
no  account  start  the  series  out  of  hand.  The  time  elapsing 
between  session  and  session,  even  if  it  is  no  more  than  24  hours, 
is  long  enough  for  O  to  get  *  rusty ' ;  only  after  a  practice  series 
does  he  warm  up  to  the  work,  as  one  says,  or  get  into  the  swing 
of  the  experiments.     This  rule  must  never  be  broken. 

As  soon  as  the  conditions  of  the  experiment  have  been 
arranged,  E  makes  out  a  plan  or  Table  in  which  all  the  con- 
stants of  the  apparatus  (distances,  heights,  etc.)  are  accurately 
entered.  The  plan  is  rigorously  adhered  to  in  the  successive 
series. 

EXPERIMENT  XIV 

Determination  of  the  DL  for  Tone. — This  experiment  is  best 
performed  by  the  aid  of  tuning  forks.  Two  arrangements  are 
in  general  use. 

( I )  Materials. — Two  forks,  the  one  of  which  is  furnished 
with      riding     weights.     Piano     hammer.     Felt.     Stop-watch. 


68 


The  Metric  Methods 


Fig.  20. 


Soundless  metronome.     [The   forks    are    so  constructed  as  to 
sound  in  unison  when  the  riders  are  set  a  certain  distance  down 

the  tines  of  the  *  variable ' 
fork  ;  this  distance  is  marked 
by  a  light  cross-cut  on  the 
tines.  The  one  rider  carries 
a  horizontal  pointer,  the  other 
a  scale.] 

(2)  Materials.  —  Two- 
forks,  each  furnished  with  a 
steel  screw,  sunk  vertically 
into  one  of  its  tines.  Piano 
hammer.  Felt.  Stop-watch. 
Soundless  metronome.  [The 
screw  is  fixed  by  a  thumb- 
nut,  and  is  turned  by  its  milled  head.  Just  below  the  milling 
of  the  head  is  a  scale,  graduated  in  tenths  of  a  whole  turn.] 

General  Directions. — It  is  clear  that  if  the  riding  weights 
are  lowered,  or  the  screw  turned  down,  the  pitch  of  the  fork  is 
raised ;  if  the  weights  are  raised,  or  the  screw  turned  up,  the 
pitch  is  lowered.  Hence  it  is  possible,  while  the  one  fork  re- 
mains constant  in  tone,  to  vary  the  pitch  of  the  other,  up  or 
down,  as  the  method  requires. 

Equality  of  the  two  vibration-rates  in  (i)  is 
given  by  the  position  of  the  cross-mark  upon  the 
tines  of  the  variable  fork:  though  the  setting  of  the 
riders  should  be  tested  (by  listening  for  beats)  be- 
fore the  experiment  is  begun.  Equality  of  the 
vibration-rates  in  (2)  must  be  determined  by  ear. 
The  screw  of  the  one  (the  constant)  fork  is 
turned  down  for  about  half  its  length.  The  screw 
of  the  other  (the  variable)  fork  is  then  gradually 
turned  down  until  beats  disappear.  "When  this 
point  has  been  reached,  comparative  tests  are 
made,  (« )  by  turning  the  screw  a  certain  number 
(say,  4)  of  whole  turns  up,  and  then  {b)  by  turning 
it  the  same  number  of  whole  turns  down,  below 
the  supposed  point  of  equality  ;  and  by  counting 
the  beats  in  the  two  cases(io  sec.  by  the  stop-watch).  If  the  point  first  determined 
was  really  the  point  of  equality  of  vibration-rates,  the  number  of  beats  in  both 
tests  will  be  the  same. 

As  regards  the  taking  of  daily  practice  series,  and  the  making 


Fig.  21. 


§    1 6.    The  Methods  of  Limits  69 

out  of  a  plan  or  Table  of  the  disposition  of  apparatus,  the  same 
rules  obtain  for  this  experiment  as  for  Exp.   XIII. 

The  course  of  the  single  observation  will  be  as  follows.  E 
says  '  Now ! '  and  after  the  usual  interval  strikes  the  first  (stand- 
ard or  variable)  fork.  When  the  fork  has  sounded,  say,  for  3 
sec,  he  damps  it  with  the  felt,  and  without  further  signal  but 
after  the  regular  interval  strikes  the  second  (variable  or  stand- 
ard) fork.  This  is  damped  in  its  turn  at  the  end  of  3  sec.  O 
then  reports  or  records  his  judgment. 

Questions. — E  and  O  {i)  What  are  the  main  characteristics 
of  the  Method  of  Limits }  What  is  the  psychology  of  the  method  i^ 

(2)  Does  the  method  presuppose  a  certain  method  of  proced- 
ure, i.  e.,  the  procedure  with  knowledge  or  the  procedure  without 
knowledge } 

(3)  Write  out  a  list  of  the  errors  to  which  the  method  is  sub- 
ject, indicating  in  brief  the  manner  of  their  avoidance. 

(4)  Make  a  diagram,  showing  the  course  of  the  method  and 
the  way  of  calculating  the  DL. 

(5)  Does  the  method  allow  of  the  interpolation  of  a  blank 
experiment .?  Suppose,  e.g.,  that  E  observes  in  6^  a  slight  ten- 
dency towards  the  error  of  bias  :  may  he  try  to  check  this  ten- 
dency by  the  introduction  of  a  blank  experiment  in  the  series } 

(6)  What  opportunity  for  introspection  does  the  course  of  the 
method  afford } 

(7)  Suggest  modifications  of  the  method. 

(8)  Discuss  the  physiology  and  psychology  of  the  constant 
error  of  space. 

(9)  Give  a  full  explanation  of  the  formulae  on  p.  65. 

(10)  Are  the  four  variable  errors  (practice,  fatigue,  expec- 
tation, habituation),  all  on  the  same  psychological  level .?  Why 
is  O  warned  particularly,  on  p.  61,  against  the  errors  of  expec- 
tation and  habituation } 

(11)  Suppose  that  you  were  planning  Exp.  XIII.  or  Exp. 
XIV.,  not  as  a  laboratory  exercise,  but  as  a  bit  of  serious  investi- 
gation :  what  improvements  or  modifications  of  technique  would 
you  suggest .? 

( 1 2)  Suggest  further  experiments  to  be  made  by  the  Method 
of  Limits. 


70  The  Metric  Methods 

§  17.  Fechner's  Method  of  Average  Error  (Method  of  Reproduc 
tion;  Method  of  Adjustment  of  Equivalent  R). — In  all  the  experi 
ments  which  we  have  hitherto  performed,  quahtative  and  quan 
titative  alike,  6^'s  part  has  been  reduced,  so  far  as  possible,  to  thai 
of  the  attentive  onlooker  ;  the  arrangement  of  apparatus  has  beer 
left,  as  exclusively  as  might  be,  in  the  hands  of  E.  In  experi 
ments  made  by  the  method  of  average  error,  this  relation  of  E 
and  O  is  changed.  E  plays  an  entirely  subordinate  part :  al 
that  he  has  to  do  is  to  prepare  the  apparatus  for  an  experiment 
and  to  record  the  value  at  which  O  sets  it.  The  setting,  the 
manipulation  and  adjustment,  are  done  by  O  for  himself. 

The  reason  for  this  change  of  relation  becomes  clear  as  soon  as 
we  state  the  problem  which  the  method  attempts  to  solve.  The 
problem  is  that  of  the  equation  of  two  stimuli.  A  certain  stimu- 
lus, r,  is  presented,  and  O  is  required  to  make  another  stimulus, 
ri,  subjectively  equal  to  it.  Let  us  take  an  instance.  The  r  mighl 
be  a  white  line  of  50  mm.  length,  shown  horizontally  upon  a  blacl^ 
background  ;  and  we  might  ask  O  to  lengthen  or  shorten  a  sim- 
ilar white  Hne,  lying  to  right  or  left  of  the  standard  r,  by  pushing 
a  vertical  black  screen  out  or  in,  until  the  two  lines  were,  in  hie 
judgment,  of  the  same  length.  E^  duties  would  then  be  confinec 
to  setting  the  movable  black  screen,  at  the  beginning  of  each 
experiment,  and  to  recording  the  length  of  7\  at  the  end. 

What,  now,  is  the  point  of  this  whole  method }  Why  is  il 
worth  our  while  to  set  such  a  task  to  O  f 

The  point  is  that,  as  we  saw  in  §  15,  the  adjustment  of  r^ 
will  never,  except  by  pure  accident,  be  objectively  right.  The 
line  which  O  judges  to  be  equal  to  r^  will,  as  a  rule,  prove  wher 
measured  to  be  longer  or  shorter  than  the  standard.  From  the 
objective  or  physical  point  of  view,  O  will  be  subject  to  errors  oi 
observation,  such  that  his  ?i  will  almost  always  differ  a  little, 
plus  or  minuSy  from  the  given  r.  These  '  errors  of  observation ' 
are,  however,  of  great  psychological  interest.  They  depend,  ir 
part,  upon  the  uncertainty  of  (9's  hand.  They  depend,  in  part,  upor 
all  those  accidental  influences,  within  or  without  6^'s  organism, 
which  lead  him  astray  in  the  particular  case, — upon  the  influences 
whose  result  we  have  discussed  in  §  15.     But  they  depend  alsc 


§    I/.  Fechners  Method  of  Average  Error  71 

upon  the  magnitude  and  variability  of  the  DL.  This  fact  alone 
recommends  them  to  our  notice,  as  students  of  quantitative 
psychology. 

It  is  in  the  *  errors  of  observation,'  therefore,  that  our  problem 
centres.  We  must  determine  them  in  such  numbers  that  we 
can  deal  with  them  systematically  and  methodically.  We  must 
seek  to  analyse  them,  and  to  refer  the  results  of  their  analysis 
to  constant  or  variable  conditions.  This  is  what  the  *  method  of 
average  error '  sets  out  to  do. 

Imagine  that  O  is  seated  before  an  apparatus  of  the  kind 
sketched  above.  In  the  middle  of  a  vertical  black  surface  is 
stretched  a  horizontal  white  line  of  120  mm.  A  fine  black 
thread,  placed  vertically,  cuts  off  50  mm.  at  the  left  of  the  line. 
We  have,  then,  two  continuous  white  lines,  the  one  of  50,  the 
other  of  70  mm.  On  the  right-hand  side  of  the  apparatus  is  a 
movable  black  screen,  of  the  same  quahty  as  the  background. 
This  screen  runs  in  horizontal  grooves,  above  and  below,  and 
can  be  pushed  in  or  out,  with  uniform  movement,  by  means  of  a 
crank  placed  conveniently  to  (9's  hand.  The  right-hand  line  can, 
therefore,  be  lengthened  or  shortened  at  pleasure.  From  the 
sides  of  the  black  background  projects  a  light  framework, — two 
outstanding  arms  carrying  a  horizontal  cross-piece, — so  con- 
structed that  O^  seated  at  a  comfortable  distance  for  vision  and 
for  handling  the  crank,  may  rest  his  forehead  against  the  cross- 
bar  during  observations.  A  mm.  scale  is  attached  to  the  fixed 
background,  so  that  movement  of  the  smaller  screen,  in  or  out, 
can  be  accurately  measured. 

We  now  have  our  conditions  ready  for  an  experiment.  E 
sets  the  movable  screen  at  such  a  point  on  the  70  mm.  line  that 
this,  the  right-hand  stimulus,  is  sensibly  larger  or  smaller  for  O 
than  the  standard  r  to  the  left.  O  settles  himself  in  position, 
takes  the  crank  in  his  hand,  and  proceeds  to  turn  the  movable 
screen  inwards  or  outwards,  towards  subjective  equaHty  of  the 
two  lines.  His  problem  is  to  set  n  at  the  length  that  best  satis- 
fies him  of  its  equality  to  the  standard  r.  Hence  he  does  not  ar- 
rest the  crank  at  the  first  point  of  apparent  equality.  Having 
reached  this  point,  he  still  continues,  very  slowly,  to  shorten  or 


72  The  Metric  Methods 

lengthen  ri.  If  he  goes  too  far,  as  he  probably  will,  he  may 
move  the  screen  out  or  in  again ;  and  he  may  continue  in  this 
way,  moving  the  screen  to  and  fro  within  narrow  limits,  until  he 
hits  upon  the  length  of  r^  which  entirely  satisfies  him.  This 
value  of  ri  is  recorded  by  E^  who  also  sets  the  apparatus  for  an- 
other experiment. 

The  experiment  is  to  be  repeated  loo  times.  The  method 
does  not  lay  down  any  rule  for  E^  initial  setting  of  r^.  It  will, 
however,  be  advisable,  on  general  principles,  (i)  to  begin  50  ex- 
periments with  Tj  >  r,  and  50  with  r^  <  r ;  (2)  to  vary  the  initial 
difference  between  r  and  r^  , — always,  of  course,  under  the  re- 
striction that  the  difference  is  clear  to  6^  at  the  outset ;  and  (3) 
to  see  to  it  that  *  very  large  '  initial  differences  occur  as  often  as 
*  very  small '  differences  on  both  sides  ( +  and  — )  of  r. 

The  100  recorded  r^  will,  as  we  have  said,  show  in  the  great 
majority  of  cases  some  difference,  plus  or  minus,  from  the  stand- 
ard r.  The  first  thing  to  do  is,  evidently,  to  determine  the  most 
probable  value  of  r^,  that  is,  the  r^  which  best  represents  0's> 
idea  (under  the  conditions  of  the  experiment)  of  the  distance  50 
mm.     The  formula  for  the  most  probable  value  is,  as  we  know, 

the  formula  for  the  arithmetical  mean :  ?!}.     We  calculate  the 

n 

mean  value  of  the  100  n,  and  enter  the  result  in  ourrecord^ 
under  the  title  Vm. 

The  differences  between  the  single  r^  and  this  r^,  taken  with- 
out regard  to  sign,  may  now  be  considered  as  O's  errors  of  ob- 
servation. Or,  in  other  words,  the  J/ F  or  AD  of  r^  affords  a  meas- 
ure of  (9's  precision.  He  has  had  r^  in  mind,  so  to  say,  as  the 
equivalent  of  r.  He  has,  on  the  average,  struck  it ;  but  in  the  single 
case  he  has  struck  a  little  to  the  one  or  the  other  side  of  it. 
The  J/ F  tells  us  how  accurate  or  inaccurate,  on  the  average, 
his  adjustments  have  been.  We  determine  it  in  the  usual  way, 
by  summing  up  the  differences  found  between  r^  and  the  separ- 
ate n,  and  dividing  the  sum  by  100,  the  number  of  experiments. 
Let  e  (error)  stand  for  these  differences.     Then  our  formula  is 

€^  =  -1.  The  value  ^m  is  known  as  the  'average  variable  error.' 
It  is  the  most  characteristic  test-value  furnished  by  the  method. 


§    I/,  Fechners  Method  of  Average  Error  73 

It  is  natural  that  we  should  bring  this  e^  (or  the  corresponding  PE^  into 
relation  with  the  MV{ox  PE{)  of  the  method  of  limits.  The  two  measures 
of  precision  cannot,  however,  be  directly  paralleled.  In  the  first  place,  the 
MV  of  the  method  of  limits  marks  the  constancy  of  the  point  at  which  O's 
judgment  changes  under  the  influence  of  a  changing  Rj  the  average  varia- 
ble error  marks  the  constancy  of  the  point  at  which  O  finally  settles  down 
to  a  judgment  of  equality.  In  the  second  place,  the  MV  of  the  method  of 
limits  expresses  the  accuracy  of  observation  only ;  the  e^  expresses  the  re- 
sultant accuracy  of  observation  and  adjustment.  In  the  third  place,  the  MV 
of  the  method  of  limits  characterises  the  accuracy  of  O's  determination  of 
the  DL;  the  e^  of  average  error  is  determined,  in  part,  by  the  magnitude 
and  variability  of  the  DL  itself.  Hence,  although  the  two  measures  belong 
to  the  same  general  class  of  test-values,  we  cannot  expect  them  to  be  numer- 
ically the  same. 

So  far,  we  have  spoken,  not  of  Vm.  itself,  but  only  of  its  MV. 
The  value  r^  the  most  probable  value  of  ri,  may  be  regarded  as- 
free  of  accidental  errors,  since  these  are  as  likely,  in  the  long  run,, 
to  fall  on  the  plus  as  on  the  minus  side  of  r,.and  will  therefore 
cancel  out  in  the  average.  It  will,  however,  almost  certainly  be 
affected  by  a  constant  error,  if  not  by  a  complex  of  constant 
errors.  The  value  of  the  crude  constant  error  may  be  deter- 
mined by  the  formula  c  =  r^  —  r:  the  value  c  being  positive  or 
negative  according  as  rm  is  >  or  <  r. 

The  mention  of  the  constant  error  c  suggests,  however,  that 
our  programme  of  the  method  is  not  yet  complete.  We  have 
been  working,  throughout,  with  r^  to  the  right  of  r.  It  is  prob- 
able, then,  that  at  any  rate  some  part  of  the  error  c  is  due  to  the 
spatial  position  of  the  two  stimuli,  and  that  this  component  can 
be  determined  (and  therefore  ehminated)  if  our  results  are  com- 
bined with  those  of  100  similar  experiments  in  which  r-^  lies  to 
the  left  of  r.  E  accordingly  shifts  the  movable  screen  to  the 
left  of  the  apparatus  (or  the  apparatus  may  be  inverted,  or  a 
second  screen  provided),  and  sets  the  vertical  black  thread  at  a 
distance  of  50  mm.  from  the  right-hand  end  of  the  120  mm.  line. 
The  crank  is  left  in  its  original  position ;  and  the  experiment 
proceeds  as  before. 

Our  100  determinations  have  thus  become  200.  But  now  that 
we  see  what  we  have  to  do,  we  can  also  see  that  the  whole  exper- 
iment has  been  injudiciously  arranged.     It  would  be  bad  policy 


74  The  Metric  Methods 

to  take  our  200  observations  in  two  steady  series  of  100  each. 
For  one  thing,  O  would  be  unpractised  at  the  beginning,  and 
highly  practised  at  the  end,  so  that  the  results  of  the  first  and 
second  100  would  not  be  comparable.  For  another  thing,  100 
determinations  are  too  many  to  take  at  a  single  sitting  ;  O  would 
grow  fatigued.  We  shall  do  well  to  make  out  a  plan  of  work  be- 
forehand :  dividing  up  our  200  observations  into  8  sets  of  25  each, 
•and  distributing  these  8  sets  in  such  a  way  that  each  of  the  four 
experimental  arrangements  (smaller  i\  right,  greater  n  right, 
smaller  r^  left,  greater  ri  left),  and  each  of  the  differences  within 
these  four  arrangements  {r^  very  much  smaller,  much  smaller, 
distinctly  smaller,  etc.),  receives  an  approximately  equal  share  of 
practice.  The  precise  arrangement  of  the  sets  need  not  be  given 
here  ;  E  will  easily  find  a  plan  of  distribution  in  which  the  effects 
of  practice  are  properly  compensated. 

At  the  conclusion  of  the  experiment,  E  sorts  out  the  200 
results  into  two  groups  of  100  each,  which  he  arranges  in  columns 
under  the  rubrics  /  {i\  to  the  left)  and  //  {r^  to  the  right  of  r). 
The  values  r^,  e^  and  c  are  to  be  determined  separately  for  each 
group  of  100  determinations. 

We  have  said  that  the  value  Cm  is  the  most  characteristic  test-value  fur- 
nished by  the  method.  We  shall  want,  then,  to  determine  <?,„  as  accurately 
as  possible.  Now  in  any  given  setting  of  ri,  the  error  e  is  algebraically  ad- 
ded to  a  constant  error  c.  But  a  '  constant '  error  is  simply  an  error  whose 
conditions  are  constant ;  its  amount  may  vary,  quite  considerably,  from  stage 
to  stage  of  a  long  series  of  experiments.  If  our  own  constant  error  c  has 
varied  in  this  way,  its  variation  must  evidently  have  affected  the  value  of  e^ 
as  determined  above.  To  free  <?^  of  any  influence  of  this  sort,  we  must 
'  fractionate  '  the  results.  We  determine  le^  not  for  the  whole  group  of  100 
observations,  but  for  quarter-groups  of  25  observations  apiece;  and  we 

jdetermine  ^^j  not  as ,  but  as 

100 

lex  +  ^^1  +  ^^3  +  ^e^ 

100 

le 

This  value  is  more  accurate  than  the  value On  the  other  hand,  it  is 

100 

indifferent  for  the  determination  of  the  c  of  each  group  whether  we  work 
from  the  group  as  a  whole,  or  separately  from  the  quarter-groups. 

We  come  now  to  the  consideration  of  the  constant  error,  in  the 


§    I/.  FecJmers  Method  of  Average  Error  75 

interest  of  which  we  increased  our  original  100  to  200  determi- 
nations.    How  are  we  to  analyse  it  ? 

We  have  already  suggested  that  c  is  made  up,  in  part  at  least, 
of  the  regular  space  error,  q.  The  q  errors  are,  by  hypothesis, 
equal  and  opposite  ;  they  increase  or  diminish  ri,  according  as 
the  two  stimuU  are  disposed  in  this  way  or  in  that ;  but  they 
increase  and  diminish  it  by  equal  amounts.     If,  then,  these  were 

the  only  errors  involved  in  Cy  we  ought  to  find  -^^i-— ^"=r.     As  a 

rule,  the  equation  does  not  hold ;  the  average  of  the  two  Vm  turns 
out  to  be  distinctly  >  or  <  r.  Where  this  is  the  case,  c  must 
have  a  second  component,  the  sign  of  which  remains  unchanged 
throughout  the  experiment.  We  will  term  it  s.  Then  we  have : 
c^=-q-^s,  rmi-r=-g-\-s, 

where  the  subscript  i  and  11  have  the  same  meaning  as  on  p.  65 
above  (i  =  ri  to  the  left,  11  =  n  to  the  right),  and  the  signs  of  ^ 
are  so  chosen  that  a  positive  or  negative  q,  as  found  by  the  equa- 
tions, corresponds  to  a  positive  or  negative  space  error  in  Fech- 
ner's  sense  (p.  65).     From  these  equations  we  obtain: 


2  2 

2       '  2  * 

Simple  as  the  calculation  is,  it  is  well  to  work  out  the  values  of 
q  and  s  from  both  sets  of  equations.  Slips  in  addition  and  sub- 
traction are  commoner  than  one  likes  to  think ! 

There  is  still  one  more  point  to  be  considered.  We  have  been  assuming 
that  the  value  c  represents  a  constant  error.  What  right  have  we  to  make 
this  assumption  ?  Why  may  not  the  occurrence  of  c  be  due  simply  to  an 
irregular  distribution  of  the  accidental  errors  <?,  itself  due  to  a  too  small  n  ? 
—  There  are  two  ways  of  answering  these  questions.  In  the  first  place,  we 
may  have  recourse  to  fractionation.  Instead  of  determining  the  c^  of  col- 
umn I,  and  the  r,,  of  column  11,  we  may  determine  four  c  for  each  column, 
— one  for  every  25  observations.  If  the  four  c  of  each  column  all  have  the 
same  sign  (-for  —  ),  we  may  be  sure  that  c^  and  ^u  represent  true  constant 
errors.  In  the  second  place,  we  may  compare  c^  and  c^^  with  the  PE  of 
the  average  r^,  and  r^n,  which  are  also  the  PE  of  rmi  —  r  and  of  rmn  —  r. 
The  value  PE„^,  it  will  be  remembered,  gives  the  limits  within  which  it  is 


Jo  The  Metric  Methods 

an  even  chance  that  the  r^  of  another  set  of  lOO  observations,  taken 
under  exactly  the  same  conditions,  will  lie :  it  may  be  determined,  approxi- 
mately, as  =  —     ^^  MV.     If,  now,  c  is  considerably  larger  than  PEm^yvt 

V  n-\ 
may  be  certain  that  it  represents  a  true  constant  error. 

EXPERIMENT  XV 

The  Equation  of  Visual  Extents. — A  very  simple  set  of 
apparatus  for  this  experiment  may  be  constructed  as  follows. 

Materials. — Galton  bar.  Black  screen.  Head-rest.  [The 
Galton  bar  shown  in  Fig.  22  consists  of  a  hardwood  metre  stick, 
planed  smooth  upon  the  surface  which  carries  the  scale  of  inches. 
Three  sliders  of  blackened  metal  are  fitted  to  the  stick  :  that  in 


r 


|ilMriiii|iiii  ^ 


^^iiiiii|iiiiiiiii[riiiiiiii|iiiiini|    ilMiiiiiiji m|ii ii|iiiiiiiii]iiiiiiiii|mini^Mii|iMiiiiiipiiiiiiii|iiimiiimii»     ^iflii|iiiimii[iiiiiiiii[iNii|iiiiiiiiiiiii[iiiUiHi[iil|ya^|ii] 


iiiiiiiiii 


Fig.  22. 

the  centre  shows  in  front  a  fine  black  wire,  set  precisely  at  the 
vertical ;  those  at  the  ends  show  larger  surfaces,  bounded  to- 
wards the  centre  by  exactly  vertical  edges.  All  three  sliders 
may  be  clamped  at  the  back  by  set-screws,  and  are  bevelled  off 
to  allow  of  accurate  readings  from  the  mm.  scale.  The  screen 
is  a  large  wooden  screen,  arranged  to  stand  vertically  on  a  table, 
and  covered  with  black  cloth  or  cardboard,  or  painted  in  dull 
black.] 

Disposition  of  Apparatus. — The  screen  is  set  up  in  a  good 
light,  but  in  such  a  position  that  it  is  not  directly  illuminated  by 
sunlight.  Two  small  screw-eyes  are  turned  into  the  back  of  the 
metre  stick,  and  two  corresponding  screw-hooks  into  the  wooden 
screen  :  so  that  the  Galton  bar  is  held  firmly  in  the  horizontal 
position  at  a  little  distance  from  the  surface  of  the  screen.  The 
central  and  one  or  other  of  the  limiting  sliders  are  clamped  fast 
at  the  standard  distance.  O  sits,  with  his  head  in  the  head-rest, 
directly  facing  the  central  black  wire,  and  at  such  a  distance  from 
the  screen  that  he  can  conveniently  manipulate  the  movable 
slider. 

After  O  has  made  a  setting,  E  gives  one  of  the  screw-hooks  a 


§  1 8.    The  Method  of  Equivalents  77 

partial  turn,  thus  releasing  the  end  of  the  bar.  Having  swung 
the  bar  out,  and  taken  his  reading,  he  resets  the  movable  slider 
and  hangs  the  bar  back  in  position. — 

Remember  that  a  few  *  practice  '  settings  must  be  made  at  the 
beginning  of  every  session  ! 

Another  experiment,  made  with  a  somewhat  modified  form  of  the  method 
of  Average  Error,  will  be  assigned  in  §21  below. 

Questions. — E  and  O  (\)  What  are  the  main  characteris- 
tics of  this  Method  of  Average  Error  1  What  is  the  psych- 
ology of  the  method } 

(2)  What  criticisms  have  you  to  pass  upon  the  method .? 

(3)  Write  out  a  list  of  the  errors  to  which  the  method  is  sub- 
ject, indicating  the  manner  of  their  avoidance. 

(4)  Name  and  characterise  the  test-values  which  the  method 
furnishes. 

(5)  Summarise  the  procedure  of  the  method. 

(6)  Have  we,  in  the  text,  determined  e^  as  accurately  as  is 
possible  t     Give  reasons  for  your  answer. 

(7)  Discuss  the  constant  error  s. 

(8)  Suggest  modifications  of  the  method. 

(9)  Discuss  the  significance  of  the  value  ^^,  with  special  ref- 
erence to  the  interpretations  of  Fechner  and  Miiller. 

(10)  Suggest  further  experiments  by  the  method. 

§  18.  The  Method  of  Equivalents.— In  Fechner's  Method  of 
Average  Error,  a  constant  stimulus  r  is  presented,  and  O  is 
required  to  adjust  a  variable  stimulus  n  to  subjective  equality 
with  r.  Suppose,  now,  that  the  stimuli  are  applied  to  a  sense 
organ  which,  Hke  the  skin,  is  variously  sensitive  at  different 
places.  In  such  a  case,  a  new  problem  arises.  We  may  require 
O  to  adjust  an  /%  applied  at  one  part  of  the  organ,  to  subjective 
equality  with  the  standard  r,  applied  at  another  and  more  (or 
less)  sensitive  part.  The  standard  may,  e.  g.^  be  applied  to  the 
forehead,  and  the  variable  to  the  back  of  the  left  hand.  We  shall 
then  obtain  an  revalue  from  the  back  of  the  hand,  which  is 
'equivalent'  in  sensation  to  the  given  r- value  on  the  forehead. 
This  is  the  aim  of  the  Method  of  Equivalents.    The  method  is,  evi- 


78  The  Metric  Methods 

dently,  identical  with  the  Method  of  Average  Error,  except  that 
the  stimuU  are  appUed  to  different  parts  of  the  organ,  or,  in  some 
rare  cases,  to  different  organs.  The  same  method  of  calculation 
may  be  employed  in  both  cases. 

It  would,  however,  in  most  instances,  be  very  inconvenient  to 
leave  the  application  and  adjustment  of  the  stimuli  to  O.  Sup- 
pose that  he  is  working  with  the  aesthesiometric  compasses.  The 
standard  distance  is  to  be  given  on  the  forehead,  the  variable  on 
the  back  of  the  left  hand.  The  left  hand  must  lie  unmoved  upon 
the  table.  O^  working  with  the  right  hand  only,  must  first  apply 
the  standard  compasses  to  the  forehead  (and  must  apply  them, 
in  successive  trials,  always  to  the  same  place  and  always  with 
the  same  amount  of  pressure);  must  then  apply  the  variable  com- 
passes to  the  back  of  the  left  hand,  and  must  vary  ri,  to  and  fro, 
until  he  has  obtained  the  distance  that  best  satisfies  him  of  its 
equahty  to  r ;  and  must  finally  measure  off  and  record  this  value 
of  r^  The  whole  experiment  is  awkward,  and  the  results  will 
necessarily  be  affected  by  a  fatigue  error  and  by  errors  of  manip- 
ulation. Besides,  O  should,  by  rights,  keep  his  eyes  closed 
during  the  comparisons. 

Conditions  may,  no  doubt,  be  arranged,  which  are  more 
favourable  to  the  employment  of  the  principle  of  average  error ; 
but  conditions  may  easily  be  found  that  are  even  less  favourable 
than  those  we  have  sketched.  On  the  whole,  it  would  seem  bet- 
ter to  give  up  this  principle,  and  to  replace  it  by  the  principle  of  the 
method  of  limits.     The  experiment  then  takes  shape  as  follows. 

Let  A  stand  for  the  forehead,  and  B  for  the  back  of  the  hand. 
Let  the  standard  distance  be  termed  a  when  it  is  given  on  A^  b 
when  it  is  given  on  B. 

We  begin  with  the  time-order  A — B.  E  first  apphes  the  stand- 
ard <2,  and  then  the  variable  ^^,  beginning  with  a  b^  that  is 
clearly  too  large.  He  gradually  reduces  ^^,  in  successive  com- 
parisons, until  b^  first  appears ^<7.  There  the  series  stops.  E 
now  takes  b^  clearly  too  small,  and  gradually  increases  it,  until 
b^  again  appears  -^^  a.  There  the  series  stops.  The  experiment 
is  repeated  lo  times,  and  the  average  value  of  b^  is  recorded 
(with  its  MV)  as  b^. 

The  time-order  is  then  changed  to  B—A.     E  first  applies  the 


§    1 8.   The  Method  of  Equivalents  79 

variable  ^^,  and  then  the  standard  a.  The  \  and  \  determina- 
tions are  repeated  i  o  times,  and  the  average  value  of  b^  is  re- 
corded (with  its  MV)  as  ^„.     The  experiment  furnishes  us  with 

the  equivalence  a  =  — - — . 

To  test  this  result,  we  repeat  the  whole  experiment,  with  reversal 
of  standard  and  variable.  Beginning  with  the  time-order  B—A, 
E  first  applies  the  standard  b,  and  then  the  variable  a^.  The 
average  value  of  the  a^,  \  and  |,  is  recorded  (with  its  MV)  as 
«j.  Turning  to  the  time-order^— ^,  E  first  applies  the  variable 
a^y  and  then  the  standard  b.  The  average  value  of  the  a^,  \ 
and  |,  is  recorded  (with  its  MV)  as  ^„.  The  experiment  fur- 
nishes us  with  the  equivalence  b  =  — — -. 

If  the  conditions  of  practice,  etc.,  have  remained  constant,  we 

shall  find  that  the  ratio  a  :  - — ^'  =  the  ratio  - — —  :  b.      We  have 

2  2 

then  obtained  the  equivalence  required.  * 

The  constant  errors  involved  in  the  values  b^,  ^„,  and  a^,  ^„, 
are  plainly  the  time  error  /  and  the  principal  error  s.  They  may 
be  determined  by  formulae  similar  to  those  on  p.  75  above. 

EXPERniENT  XVI 

The  Method  of  Equivalents  as  applied  to  Cutaneous  Extents, 
— The  only  Materials  required  for  this  experiment  are  the  aesthe- 
siometric  compasses  used  in  Vol.  I.,  Exp.  XXXIV. 

General  Directions. — We  may  choose,  as  obvious  surfaces 
to  work  upon,  the  volar  side  of  the  wrist,  the  ball  of  the  thumb, 
and  the  tip  of  the  forefinger.  Since  the  R  with  which  he  is  to 
work  must  be  distinctly  supraliminal,  E  will,  first  of  all,  roughly 
determine  (9's  two-point  limen  at  these  places.  That  done,  he 
selects  separations  of  the  compass  points  that,  while  supraliminal, 
are  yet  not  so  large  as  to  prevent  serial  increase  within  the  limits 
of  the  area  of  stimulation.  Thus,  if  the  R  are  to  be  applied  in 
the  longitudinal  direction,  he  may  select  distances  in  the  neigh- 
bourhood of  5  mm.  for  the  finger,  10  mm.  for  the  thumb,  and  20 
mm.  for  the  wrist.     Finger  is  now  to  be  equated  with  both  thumb 


8o  The  Metric  Methods 

and  wrist,  thumb  with  both  finger  and  wrist,  etc.  The  equiva- 
lent extents  are  roughly  determined  in  preliminary  experiments, 
and  the  series  proper  then  begin. 

As  always,  a  practice  series  must  be  taken  at  the  beginning  of 
every  laboratory  session. 

EXPERIMENT  XVII 

The  Method  of  Equivalents  as  applied  to  Cutaneous  Pressure. 
— Materials. — Cartridge  weights.  Soundless  metronome. 
Stop-watch.  [The  weights  are  paper  cartridge  cases,  3  cm.  in 
height  and  2  cm.  in  diameter,  loaded  with  shot  and  stuffed  with 
cotton  wool.  The  top  is  closed  by  a  wad,  and  the  metal  bottom 
is  covered  with  a  layer  of  blotting  paper.  There  are  10  weights 
in  all,  varying  by  2  gr.  steps  from  1 2  to  30  gr.] 

General  Directions. — The  standard  weight  is  that  of  8  gr., 
applied  vertically  to  the  back  of  6>'s  hand.  E  is  to  determine  the 
equivalent  of  this  weight,  upon  the  volar  surface  of  the  opposite 
wrist.  He  first  determines  the  equivalent  roughly,  in  preliminary 
experiments,  and  then  plans  his  series.  The  weights  are  left 
upon  the  skin  for  2  sec,  and  there  is  an  interval  of  3  sec.  between 
standard  and  variable.  A  pause  of  30  sec.  is  made  after  each 
comparison,  and  a  pause  of  at  least  3  min.  between  series  and 
following  series. 

In  this  experiment  practice  easily  wears  off,  and  fatigue  easily 
sets  in.  To  guard  against  the  first  source  of  error,  the  warming-up 
series  at  the  beginning  of  each  session  must  be  extended  some- 
what beyond  the  usual  limits ;  to  guard  against  the  second, 
the  time  relations  of  the  experiment  must  be  very  strictly 
observed. 

Questions. — E  and  O  {i)  Write  out  in  full  the  procedure  of  the 
Method  of  Equivalents,  regarded  as  a  form  of  the  method  of  aver- 
age error,  showing  the  derivation  of  the  test-values. 

(2)  Criticise  the  method  (appHcation  of  the  method  of  limits) 
as  formulated  in  the  text. 

(3)  Suggest  modifications  of  the  method. 

(4)  Criticise  the  procedure  for  the  elimination  of  constant 
errors, 

(5)  Criticise  Fechner's  theory  of  a  proportional  constant  error. 


§    1 9-    The  Method  of  Equal  Sense  Distances  ^l 

(6)  Under  what  conditions  may  equivalents  be  obtained  from 
two  different  sense  departments  ?  Point  out  the  chief  sources 
of  error. 

O  {y)  What  complicating  factor  in  judgment  does  introspec- 
tion reveal  in  Exp.  XVI.  that  is  absent  in  Exp.  XVII.  ? 

§  19.  The  Method  of  Equal  Sense  Distances  (Method  of  Mean 
Gradations;  Method  of  Supraliminal  Differences ;  Method  of  Equal- 
appearing  Intervals). — The  method  which  we  are  now  to  discuss 
is  concerned  directly  with  the  measurement  of  sense  distances. 
We  have  referred  to  it  above,  pp.  25  f.,  and  we  have  had  a  rough 
illustration  of  its  use  in  Exps.  XL,  XII. 

Let  us  suppose  that  we  are  to  work  with  intensities  of  sound. 
We  require  a  *  sound  pendulum,'  such  as  is  shown  in  Fig.  23.  A 
pendulum  rod,  suspended  from  a  steel  pillar,  ends  in  a  hard-rubber 
ball  which  strikes,  as  the  pendulum  falls,  upon  a  block  of  ebony. 


Fig.  23. 

A  graduated  arc,  attached  to  the  base  of  the  instrument,  shows 
the  angle  through  which  the  pendulum  falls.  Since  the  intensity 
of  the  sound  produced  is  directly  proportional  to  the  height  of 
fall,  or  (what  in  this  case  is  the  same  thing)  to  the  square  of  the 
sine  of  half  the  arc  through  which  the  pendulum  swings,  we  can 
readily  calculate  the  relative  intensities  of  the  sounds  employed 

by  the  formula  i=  sin^  -,  where  6  is  the  angle  at  which  the  pendu- 


8  2  The  Metric  Methods 

lum  is  set  for  the  experiment.  To  ensure  accuracy  of  adjustment^ 
the  pendulum  is  not  dropped  by  hand,  but  released  by  a  mechanical 
device.  The  releases  (three  in  number)  slide  up  and  down  upon 
the  graduated  arc,  and  can  be  fixed  at  any  height ;  their  manipu- 
lation is  easy,  and  they  can  be  turned  back,  out  of  use,  when  not 
wanted.  The  sounds  must  be  bright,  clear-cut,  without  reso- 
nance. We  therefore  wind  a  piece  of  small  rubber  tubing  round 
the  pendulum  rod,  push  a  piece  of  sponge  rubber  over  the  free 
end  of  the  metal  arc,  and  set  the  base  of  the  instrument  upon 
layers  of  thick  felt.  With  these  precautions  the  pendulum  ought, 
if  it  is  well  constructed,  to  give  us  sounds  that  vary  in  intensity, 
but  show  no  differences  of  quality  and  no  after-effects. 

If  we  term  the  sound  produced  by  a  fall  through  io°  the  inten- 
sity I,  we  find  by  our  formula  that  the  sounds  produced  by  a  fall 
through  20°,  30°,  40°,  50°,  60°  are  roughly  4.0,  8.8,  15.4,  23.5, 
32.9.  Suppose  that  we  are  working  between  the  limits  30°  and 
60°.  We  have  a  stimulus  difference  of  8.8  to  32.9  ;  or,  if  we 
make  30°  our  unit,  a  difference  of  i  to  (approximately)  3|. 
Corresponding  to  this  stimulus  difference  is  a  sense  distance,  be- 
ginning with  a  relatively  weak  or  faint,  and  ending  with  a  rela- 
tively strong  or  loud  sound.  Our  problem  is,  now,  to  find  a 
stimulus  intensity  that  lies  midway,  for  sensation,  between  8.8  and 
32.9  or  between  i  and  3|.  Call  the  given  sense  distance  a  c. 
We  are  to  divide  this  into  the  two  equal  sense  distances  a  b  and  b  c. 
There  are  various  ways  in  which  we  may  seek  to  accomplish  our 
purpose. 

(i)  The  most  natural  way,  perhaps,  is  to  apply  to  our  present 
problem  the  principle  of  the  method  of  limits.  Let  r^  and  r^, 
of  which  ^3  is  the  greater,  represent  the  two  extreme  heights  of 
fall.  Then  we  choose  an  intermediate  height  of  fall,  r^  {v  = 
variable),  that  gives  a  sound  which  is  clearly  too  loud.  Taking 
the  stimuli  always  in  the  order  rg,  i\,  r^,  we  decrease  r^  by  small 
steps  until  we  reach  the  point  at  which  the  upper  sense  distance 
(the  difference  between  the  strongest  and  the  variable  stimulus) 
just  ceases  to  appear  smaller  than  the  lower  sense  distance.  The 
scale  value  is  recorded  as  r'^  (<af=  descending),  with  the  corre- 
sponding judgment  of  .?  or=.  We  do  not  stop  the  series  at  this 
point,  but  continue  to  decrease  r„,  until  we  reach  the  point  at 


§    1 9-    The  Method  of  Equal  Sense   Distances  83 

which  the  upper  sense  distance  first  appears  larger  than  the 
lower.  The  scale  value  is  recorded  as  r" ^^  with  the  corresponding 
judgment  of  >.  Here  the  \  series  ends.  Now  the  experiment  is 
reversed.  We  start  with  an  Vv  that  gives  a  sound  which  is 
clearly  too  weak,  and  (still  taking  the  three  stimuli  in  the  order  rg, 
r,„  r^  increase  r^  by  small  steps  until  we  reach  the  point  at 
which  the  lower  sense  distance  just  ceases  to  appear  smaller  than 
the  upper.  The  scale  value  is  recorded  as  r\  ((2  =  ascending), 
with  the  corresponding  judgment  of  .?  or=.  The  series  is  con- 
tinued to  the  point  at  which  the  lower  sense  distance  first  ap- 
pears larger  than  the  upper.  The  scale  value  is  recorded  as  r" a^ 
with  the  corresponding  judgment  of  >. 

The  four  values  of  r^  which  we  have  thus  obtained  are  deter- 
mined by  help  of  the  formula  on  p.  81.  Their  average  gives  us 
^2  or  the  r^  required ;  the  most  probable  value  of  the  stimulus 
which  lies  midway,  for  sensation,  between  rgand  r^,  — the  stimulus 
whose  sensation  b  bisects  the  given  sense  distance  a  c. 

So  far,  however,  we  have  worked  only  in  the  time  order  rg,  r^, 
r^.  To  eliminate  a  possible  constant  error  of  time,  we  must  now 
^ive  the  stimuli  in  the  order  ri,  r^,  7%.  We  determine  the  val- 
ues of  r'a,  r"a,  r'^,  r\y  as  before,  and  average  them  to  an  r^. 

Each  of  these  experiments  should  be  repeated  6  times.  The 
12  ^2  are  averaged  to  a  final  value,  and  the  MV  is  recorded  as  a 
measure  of  precision. 

The  final  value  found  for  r^  gives  us  an  intermediate  stimu- 
lus that  should  be  free  of  constant  errors,  and  is  as  free  of  vari- 
able errors  as  we  can  make  it  in  the  time  at  our  disposal.  Let 
its  intensity  be  x.  We  then  have  the  equation :  the  sense  dis- 
tance corresponding  to  ;ir  — 8. 8  =  the  sense  distance  corresponding 
to  32.9— ;r.  Or,  with  30°  as  our  unit  of  stimulus:  the  distance 
.;i;— i=the  distance  S^—x.  If  ^  is  the  arithmetical  mean  of 
the  two  extremes,  t.  e.,  if  in  the  last  equation  it  is  (approxi- 
mately) 2.37,  we  have  found  that  equal  sense  distances 
correspond  to  equal  differences  of  stimulus  intensity.  If  ;i:  isthe 
geometrical  mean  of  the  two  extremes,  t.  e.,  if  it  is  (approxi- 
mately) 1.9,  we  have  found  that  equal  sense  distances  correspond 
to  relatively  equal  differences  of  stimulus  intensity. 


84  The  Metric  Methods 

It  need  hardly  be  said  that  precisely  the  same  rules  must  be  followed  here 
as  were  prescribed  in  previous  experiments  for  the  method  of  limits.  The 
single  series  must  be  short.  The  starting-point  of  the  series  should  be 
varied ;  but  every  4'  series  should  be  matched  by  an  f  series  of  approxi- 
mately the  same  length.  The  size  of  the  steps  should  be  kept  constant  within 
each  paired  series.     And  so  on. 

We  must  take  care  that  practice  is  evenly  distributed,  both  over  the  two 
directions  (i,  -f)  and  over  the  two  time  orders.  The  schema  of  an  experi- 
ment  thus  takes  shape  somewhat  as  follows  : 


Series                          Time  order 

Direction 

I                                  rz  r^  ^1 

1 

2                                  ^3  r^  n 

t 

3                                                     ^1    ^v    ^3 

t 

4                                                  ^1    ^v    ^3 

>^ 

5                               ^1  ^v  H 

\ 

6                                  ri  r^  rg 

\ 

7                                H  r^  ^1 

\ 

8>                                 r^  r^  n 

t 

Three  experiments  of  this  sort  furnish  the  1 2 

required 

determinations  of  n. 

(2)  We  may  also  employ  the  method  of  limits  in  its  alternative 
form.  The  essential  point  in  this  procedure  is  that  r^  is  not  varied 
continuously  in  one  direction,  but  is  taken  at  haphazard,  now- 
weak  now  strong,  now  nearer  r^  or  nearer  r^^  now  precisely  mid- 
way (arithmetically  or  geometrically)  between  r^  and  r^.  O 
never  knows  what  to  expect,  and  cannot,  therefore,  be  biassed  or 
expectant.  The  sounds  may  come  in  the  order  fg,  /^,,  r^  or  in  the 
order  r^,  r^,  rg,  and  r^  may  lie  at  any  point  between  the  extremes. 
In  the  particular  case,  O  has  to  judge  of  r^  as  too  high,  too  low, 
or  middle :  the  judgment  t  is  also  allowed. 

E  accordingly  makes  out  a  plan  of  experimentation  beforehand. 
He  first  determines  the  limits  within  which  r^  shall  be  varied.  No 
rule  can  be  given,  except  that  it  is  better  to  take  the  limits  too 
wide  than  too  narrow.  He  next  determines  the  size  of  his  steps. 
These  may  be  wider  toward  the  limiting  values  of  r^,  narrower 
toward  the  region  of  the  means.  Like  the  values  of  r^  itself, 
they  should  be  disposed  symmetrically  about  this  region.  Then 
he  determines  the  number  of  experiments  to  be  made  with  each 
value  of  r^.  He  need  make  but  few  (say,  5)  with  the  outlying 
values ;  he  must  make  more  (say,  10)  in  the  middle  region,  i,  e.^ 


§   19-    The  Method  of  Equal  Sense  Distances  85 

from  a  point  somewhat  below  the  geometrical  to  a  point  some- 
what above  the  arithmetical  mean.  Finally,  he  makes  out  his 
haphazard  series.  Their  arrangement  is  not  to  be  left  wholly  to 
chance  :  for  it  is  important  {a)  that  the  time  order  (r^,  r„,  rg,  and 
Tg,  r^,  r^  be  frequently  changed ;  {U)  that  the  values  of  r^  in  two  suc- 
cessive observations  be  not  too  near  together  ;  and  {c)  that  the 
arrangement  of  the  observations  in  any  single  series  be  exactly 
reversed  in  some  subsequent  series.  The  first  two  rules  explain 
themselves  ;  the  object  of  the  third  is  to  compensate  the  possible 
influence  upon  judgment  of  the  observation  (or  observations) 
immediately  preceding.  These  directions  are  more  formidable  in 
statement  than  they  are  in  actual  work. 

The  results  are  thrown  into  tabular  form.  The  first  column 
of  the  Table  shows  the  values  of  r^  employed,  from  the  lowest  to 
the  highest,  as  calculated  by  the  formula  on  p.  8 1 .  The  next 
three  columns  give,  under  the  general  rubric  \  (time  order  r^, 
r^y  Tg),  the  number  of  judgments  for  I  {r^  too  low),  m  {r^  in  the 
middle,  or  the  relation  of  the  two  sense  distances  doubtful) 
and  h  [r^  too  high).  The  next  three  columns  give,  similarly, 
under  the  general  rubric  \  (order  ^g,  r^,  r^),  the  number  of 
judgments  for  /,  m  and  h.  A  final  column,  headed  n^  shows  the 
number  of  observations  taken  with  each  value  of  r^. 

The  first  thing  to  do,  by  way  of  calculation,  is  to  halve  the  m 
cases  between  the  /  and  h  cases  of  the  same  horizontal  line.     We 

thus  obtain  I'  =  l-{.—,  h'  =  h-\-—.     The  proceeding  is  justifiable 

on  the  ground  that  the  difference  between  an  m  and  an  h  ox  I 
judgment  is  quantitative  only,  not  qualitative;  the  just  discrimi- 
nable  /  and  h  cases  are  less  different  from  the  nt  cases  than  they 
are  from  one  another.  The  arithmetical  means  of  the  h'  and  /' 
under  the  two  rubrics  \  and  |  are  then  determined,  and  ex- 
pressed as  percentages.  We  thus  get  a  second  Table,  of  three 
columns :  the  first  gives,  as  before,  the  values  of  r^,  the  second 
the  percentages  of  /'  judgments,  and  the  third  the  percentages 
of  h'  judgments  corresponding  to  these  values. 

The  subjective  mean  between  r^  and  r^  lies,  evidently,  at  the 
value  of  r^  for  which  l'  =  h'=^o%.  It  is  not  likely  that  we 
shall  have  struck  this  precise  point  in  our  experiments.     In  all 


86  The  Metric  Methods 

probability,  we  shall  find,  for  two  successive  values  of  r^  in  the 
central  region,  some  such  relation  as  this : 

I'  h! 

rv=x  52.5  47.5 

rv=y  40.0  60.0, 

from  which  we  can  see  that  the  subjective  mean  between  i\  and 
r^  is  an  r^  that  lies  between  x  and  j,  but  cannot  tell  precisely 
what  value  should  be  assigned  to  r^.  If,  now,  we  term  the  I' 
respectively  I' ^  and  Z^,  and  the  h'  similarly  h' ^  and  h' ^^  we  can 
determine  the  required  i\  by  the  formula : 


n  = 


I'  X  —  i-  y  '^'-  y' 


We  thus  obtain,  though  by  a  different  procedure,  the  same  prin- 
cipal test-value  that  was  afforded  by  the  first  form  of  the  method. 
The  second  form,  unfortunately,  does  not  yield  a  measure  of  pre- 
cision, corresponding  to  the  MV. 

It  was  suggested  above  that  10  observations  be  taken  in  the  central,  and 
5  in  the  outlying  regions.  If  time  allow,  the  experiment  should  be  con- 
tinued until  these  numbers  have  been  raised  to  50  and  25  respectively. 

(3)  We  said  at  the  outset  that  there  were  various  ways  of  at- 
tacking the  problem  of  this  §.  Two  of  these  we  have  now  dis- 
cussed. Another,  though  only  a  preliminary  method  is,  as  was 
remarked  at  the  beginning  of  the  §,  the  method  of  group  arrange- 
ment which  we  employed  in  Exps.  XL,  XII.  By  this  method 
we  are  able  to  determine  not  only  the  values  of  the  various  inter- 
mediate Ry  but  also  the  MV oi  each  group. 

If  we  do  not  possess  the  instruments  required  for  work  with  the 
method  of  limits,  we  may  equate  two  sense  distances,  very  sim- 
ply, by  a  method  of  adjustment,  i.  e.,  by  a  method  akin  to  the 
method  of  average  error.  We  may,  e.  g.,  take  squares  of  black 
and  white  cardboard,  and  seek  in  a  series  of  trials  to  paint  with 
Indian  ink  a  grey  that  shall  lie  midway  for  sensation  between  the 
two  extremes  of  brightness.  The  photometric  values  of  the 
middle  greys  can  then  be  determined  (either  directly,  or  after 
equation  to  a  grey  disc)  by  the  Kirschmann  photometer.  The 
simplicity  of  such  a  method  is,  however,  offset  by  its  roughness 
and  unreliability. 


§    1 9-    The  Method  of  Equal  Sense   Distances  8/ 

Finally,  we  may  use  for  the  determination  of  r^  a  form  of  the 
method  of  constant  i^-differences  which  we  discuss  in  §  22. 
This  method  furnishes  a  measure  of  precision. 

EXPERIMENT    XVHI 

The  Application  of  the  Method  of  Equal  Sense  Distances  to 
Intensities  of  Sound. — Materials. — Sound  pendulum.  Sound- 
less metronome. 

General  Directions. — E  must  first  of  all  select  his  two  lim- 
iting sensations,  above  and  below.  The  interval  between  them 
should  be  as  wide  as  possible.  At  the  same  time,  they  must 
appeal  to  (9  as  '  going  together,'  as  constituting  points  upon  one 
and  the  same  sound  scale.  If  the  weaker  sound  seems  flat,,  dull, 
dead,  *  curiously '  weak  :  or  if  the  louder  sound  seems  surpris- 
ingly, startlingly  loud  or  emphatic  :  the  limits  have,  for  this  O  at 
this  stage  of  practice,  been  drawn  too  widely. 

When  the  limiting  intensities  have  been  chosen,  preliminary 
experiments  are  made  to  determine  the  region  of  the  sound 
scale  within  which,  for  (9's  sensation,  the  mid-intensity  lies. 
The  result  will  be  quite  rough. 

E  may  then  proceed  to  the  regular  series.  O  sits  with  closed 
eyes,  about  i  m.  from  the  pendulum,  and  gives  his  maximal 
attention  to  each  triple  impression  as  it  comes.  The  three  R 
are  sounded  at  1.5  sec.  intervals  (measured  by  the  metronome). 
The  interval  between  observation  and  observation  will  depend 
upon  ^'s  handiness  with  the  instrument :  it  should  be  kept  con- 
stant.    A  pause  of  some  min.  is  made  after  each  series. 

Practice  experiments,  for  warming  up,  must  be  taken  at  the 
beginning  of  every  laboratory  session. 

If  time  permits,  both  forms  of  the  method  should  be  em- 
ployed. 

EXPEREVIENT  XIX 

The  Application  of  the  Method  of  Equal  Sense  Distances  to 
Brightnesses. — There  are  various  ways  of  performing  this  experi- 
ment. We  will  first  of  all  describe  the  manner  in  which  it  was 
originally  made  by  Delboeuf. 

Materials. — (i)  Delboeuf  disc.     Colour  mixer.    Metal  pro- 


88 


The  Metric  Methods 


Fig.  24. 


tractor.  Dark  box.  Head-rest.  [The  Delboeuf  disc,  in  its  first 
form,  is  shown  in  Fig.  24.  The  centre  piece  A^  10  cm.  in  diam., 
is  composed  of  two  discs  of  cardboard  with  a  cardboard  star  of 

the  same  diam.  pasted  be- 
tween them  ;  its  edges  pre- 
sent a  series  of  slits,  in 
which  the  sectors  ^,  ^,  Cy 
etc.,  may  be  inserted.  The 
face  of  A  is  covered  with 
black  velvet.  The  radii 
of  the  white  cardboard 
sectors  a^b^  Cy  etc.,  increase, 
from  A  outwards,  in  steps 
of  23  mm.;  so  that  the  ex- 
treme diameter  of  the  com- 
pound disc  is  23.8  cm. 
The  sectors  themselves 
have  their  degree-values 
marked  on  their  backs,  and  a  sufficient  number  of  them  is  cut  to 
produce  all  the  arcs  between  1°  and  360°  of  white. 

It  is,  however,  not  worth  while  to  reproduce  the  Delboeuf  disc 
in  this  particular  form.  A  great  many  sectors  must  be  cut ;  the 
sectors  become  worn  and  dirty  from  much  handling  ;  and,  at  the 
best,  a  sector  is  likely  to  fly  out  of  A  and  get  broken  during  the 
course  of  a  series.  It  is  better  to  cut  in  one  piece  a  cardboard 
disc  composed  of  A  and  of  the  a  and  c  sectors,  and  in  another 
piece  a  triple  b  sector,  and  to  mount  the  latter  behind  the  former 
on  the  colour  mixer.  We  then  have,  if  A  be  covered  with  black 
and  the  compound  disc  rotated  before  a  dark  chamber,  three 
grey  rings,  the  inner  and  outer  of  which  (corresponding  to 
the  a  and  c  sectors)  are  constant,  while  the  middlemost  (corre- 
sponding to  the  b  sectors)  is  variable,  according  as  the  triple  b 
sector  is  pushed  behind  the  c  sectors  or  drawn  out  from  them. 
Here  are  the  conditions  for  work  with  the  method  of  equal 
sense  distances. — We  must,  of  course,  have  more  than  one  of 
the  Aac  discs,  if  we  are  to  vary  the  limits  between  which  the  ex- 
periment is  taken. 

Delboeuf s  dark  box  is  shown  in  Fig.  25.     It  consists  of  a 


§   ig.    The  Method  of  Equal  Sense  Distances  89 

blackened  wooden  box  ABCD,  65x50x50  cm.,  at  the  back  of 
which  is  placed  obliquely  a  board  CE  faced  with  black  velvet. 
The  box  is  so  disposed  that  the  light  from  the  window  at  L 


y 


y 


y 


/ 


/ 


./ 


B. 

E' 

f<y 

/ 

,'' 

/ 

• 

/ 

^ 

• 

X 

,y 

/'"^'' 

/ 

.^'-"HC". 

/y 

/y 

/' 

^^, '-&' 


Fig.  25. 

does  not  fall  directly  upon  CE.  The  disc  stands  in  the  opening 
of  the  box,  with  its  centre  at  M ;  the  observer  is  at  O. — The  tube 
of  the  Kirschmann  photometer  will  answer  the  same  purpose  as 
this  dark  box.] 

(2)  Delboeuf  disc.  Colour  mixer.  Protractor.  Kirschmann 
photometer.  Head-rest.  [The  Delboeuf  disc  may  also  be  con- 
structed as  shown  in  Fig. 
26.  Upon  a  disc  of  white 
cardboard  are  pasted  two 
black  sectors,  corresponding 
to  a  and  c  of  Fig.  24.  Over 
the  disc  is  laid  a  separate 
black  sector,  corresponding 
to  b  of  the  former  Fig. 
On  rotation,  three  grey 
rings  appear ;  the  middle- 
most may  be  varied.] 

(3)  Three  colour  mixers. 
Six  small  discs,  three  black 

and  three  white.  Three  backgrounds,  two  constant  and  one  vari- 
able.    Head-rest  with  observation  tube.    Protractor.    Kirschmann 


Fig.  26. 

The  middle  sector  is  shown  both  separately 

and  in  position. 


90  The  Metric  Methods 

photometer.  [If  one  were  asked  to  suggest  a  procedure  for  this 
experiment,  it  would  perhaps  seem  most  obvious  to  take  three 
mixers ;  to  mount  upon  each  one  a  compound  disc  of  black  and 
white ;  to  stand  the  three  mixers  side  by  side  in  the  same  straight 
line ;  and  then  to  set  the  right  and  left  hand  greys  at  a  constant 
value,  and  to  vary  the  middle  grey  as  the  method  requires.  The 
difficulty  would,  however,  at  once  arise  that  the  three  greys 
must  be  seen  against  a  background  ;  and  that,  if  the  background  is 
the  same  for  all,  contrast  effects  are  produced  which  vary  from 
disc  to  disc  and  are  not  eliminable  by  any  change  of  position.  It 
is  necessary,  therefore,  that  each  disc  be  viewed  against  a  back- 
ground of  its  own  brightness.  An  arrangement  that  fulfils  this 
condition  is    shown    in  Fig.  2J.    The   discs   are   placed  in  the 

V 


Fig.  27. 

order  darker  (^),  variable  {v)^  lighter  (/).  Behind  d  and  /  stand 
constant  backgrounds  c  of  grey  paper,  matched  to  the  brightness 
of  the  discs.  Behind  v  stands  the  large  compound  disc  V,  which 
is  varied  as  v  is  varied.  In  another  arrangement,  recourse  is  had 
to  the  triple  mixer  shown  in  Fig.  28.  The  three  large  discs  serve 
as  the  backgrounds,  before  which  the  mixers  that  carry  the 
small  discs  are  set  up.] 


Fig.  28. 


General  Directions. — As  regards  choice  of  limits,  prelimi- 
nary experiments,  and  warming-up  series,  the  same  rules  obtain 
for  this  exp.  as  for  Exp.  XVIII.  Note  that  the  time  error  of 
Exp.  XVIII  is  here  replaced  by  a  space  error.     We  cannot  elim- 


§    19-    The  Method  of  Equal  Sense  Distances  91 

inate  this  error  with  the  disc  employed  in  ( i ),  because  we  cannot 
put  the  lightest  ring  on  the  outside  ;  we  can,  however,  by  a  slight 
modification,  bring  the  darkest  ring  to  the  periphery  of  the  disc 
used  in  (2) ;  and  we  can  interchange  the  extreme  discs  of  ar- 
rangement (3). 

(i)  One  of  the  three  sectors  of  the  triple  piece  b  is  graduated  at  the  back 
in  0.5°  units.  The  unit  of  serial  change  is  thus  1.5°.  The  final  reading,  at 
the  end  of  a  series,  is  taken  with  the  protractor. 

(2)  If  we  regard  the  black  of  the  Uelboeuf  box  or  the  Kirschmann  tube 
as  representing  the  o  of  light  intensity,  we  may  report  the  result  of  (i)  ia 
degrees  of  white.  If,  however,  we  employ  the  second  form  of  the  Delboeuf 
disc,  or  the  three  mixers  of  (3),  we  must  reduce  our  results  to  their  photo- 
metric values  by  aid  of  the  Kirschmann  photometer:  see  p.  37  above. 

(3)  It  is  very  important  that  the  discs  and  backgrounds  of  this  experiment 
be  uniformly  lighted.  Hence  one  ha?  practically  no  choice  but  to  work  in 
the  dark  room.  O  is  seated  at  a  low  table  facing  the  discs,  and  observes  them 
through  a  truncated  cone  of  black  cardboard.  The  source  of  illumination 
{e.g.^  a  row  of  Welsbach  burners)  is  then  placed  before  a  white  screen  upon 
a  stand  or  table  erected  above  that  at  which  O  sits. 

The  triple  mixer  of  Fig.  28  may  be  run  by  an  electric  motor  placed  in  an 
adjoining  room. 

In  the  observations  of  (i)  and  (2),  O  sits  at  a  distance  of  about 
I  m.  from  the  disc.  The  distance  of  O  from  the  three  discs  of 
(3)  must  be  regulated  by  circumstances. 

If  time  permits,  both  forms  of  the  method  should  be  employed. 

Questions. — E  and  O  {\)  Discuss  the  relative  advantages 
and  disadvantages  of  the  two  forms  of  the  Method  of  Limits,  as 
employed  for  the  equation  of  two  sense  distances. 

(2)  What  are  the  principal  sources  of  error  in  the  Method  of 
Equal  Distances  } 

(3)  Discuss  the  validity  of  rule  {c),  p.  85. 

(4)  Criticise  the  method  of  reduction  of  /,  m,  h  to  l\  h\ 

(5)  Criticise  the  formula,  p.  86. 

(6)  The  second  form  of  the  method  of  limits  does  not  yield  a 
measure  of  the  precision  of  i\.  Does  it  tell  us  anything  at  all  of 
the  variability  of  this  value } 

(7)  Discuss  the  means  proposed  for  the  elimination  of  the  con- 
stant error  of  time. 

(8)  Suggest  modifications  of  the  method. 


92  The  Metric  Methods 

(9)  Give  a  full  introspective  account  of  your  method  of  esti- 
mating the  two  sense  distances  presented. 

(10)  How  would  you  change  the  disc  of  Exp.  XIX.  (2)  for 
•elimination  of  the  space  error } 

(11)  Is  there  any  specific  source  of  error  in  Exp.  XIX.  that 
does  not  affect  the  results  of  Exp.  XVIII..? 

(12)  Which  of  these  exps.  is  the  easier  for  O  ?     Why  .? 
{13)  Suggest  further  experiments  by  the  Method. 

§  20.  The  Method  of  Constant  Stimuli  (Method  of  Right  and 
Wrong  Cases). — The  method  which  we  are  now  to  discuss,  is,  in 
a  way,  the  direct  opposite  of  the  Method  of  Limits.  In  seeking 
to  determine  an  RL  by  that  method,  we  vary  the  R  until  we  ob- 
tain a  certain  limiting  judgment,  the  form  of  which  we  have  set- 
tled beforehand.  In  seeking  to  determine  an  RL  by  the  Method 
of  Constant  Stimuli,  we  keep  the  R  the  same,  throughout  the 
experiment,  and  let  the  judgments  vary  as  they  will  under  the 
influence  of  variable  and  accidental  errors.  At  the  end  of  the 
experiment,  we  group  the  judgments  into  classes,  and  from  the 
law  of  their  distribution  (the  relative  number  in  each  class)  ascer- 
tain, by  mathematical  means,  the  amount  of  R  which  corresponds 
to  the  RL  required.  In  the  one  method,  then,  stimuU  vary  and 
judgments  are  constant ;  in  the  other,  judgments  vary  and 
stimuli  are  constant. 

We  may  illustrate  the^  procedure  by  reference  to  work  in 
aesthesiometry.  Suppose  that  we  wish  to  determine  the  *  space 
limen,'  or  limen  of  dual  impression,  for  the  lower  eyelid.  The 
first  thing  to  do  is  to  make  out  a  graduated  series  of  stimuli,  i.  e.y 
of  *  distances,'  or  separations  of  the  compass-points.  The  series 
begins  with  a  distance  /)  =  o  ;  that  is  to  say,  only  one  compass-point 
is  employed.  It  ends  with  a  Z)  so  large  that  O  always  (or  nearly 
always)  gives  the  judgment  *  two  points '  when  the  stimulus  is 
applied.  The  value  of  this  upper  D  must  be  ascertained  by  pre- 
liminary experiments.  In  the  particular  case  before  us,  the 
series  has  been  worked  out  as  follows : 

456 
9      Hi     i3f 


jC^in  Paris  lines      0 

0.5 

I 

1-5 

2 

3 

D  in  mm.  (approx- 

imately)             0 

H 

H 

31 

4* 

6f 

§  20.    The  Method  of  Constant  Stimuli  93 

Other  series  would  do  as  well :  we  might  take,  e.  g.,  o,  1,2,  3, 
4,  6,  8,  10,  12  mm.  The  series  quoted  has,  however,  been  actually 
employed. 

Each  one  of  these  9  stimuli,  mixed  in  haphazard  order,  is  to 
be  set  down  upon  the  skin,  say,  100  times.  (9's  judgment  of  the 
same  stimulus  will  vary,  from  one  observation  to  another,  as  the 
influence  of  the  accidental  errors  varies.  Three  forms  of  judg- 
ment are  open  to  him  :  '  two  points,'  *  one  point,'  *  doubtful ; ' 
they  are  entered  upon  his  record  sheet  as  2,  i,  .?.  At  the  end  of 
the  experiment,  E  determines  for  each  stimulus  the  relative  num- 
ber or  percentage  of  cases  in  which  these  three  judgments  have 
occurred.  Thus,  in  the  investigation  already  cited,  the  relative 
number  of  two-point  judgments  was  as  follows  : 

Dm  Paris  lines       o      0.5       i       1.5       2        3        4        5         6 
Two-point  judg- 
ments 0.30  o.io  0.14  0.40  0.65  0.80  0.87  0.96  1. 00. 

This  means,  e.  g.,  that  with  a  Z>  of  3  Paris  lines  O  said 
^Two'  in  80%  of  the  observations,  while  in  the  remaining  20%  he 
said  *  One '  or  '  Doubtful.' 

These  are  the  results  that  the  Method  of  Constant  Stimuli  fur- 
nishes. We  have  now  to  find  some  way  of  turning  them  to 
account  for  a  determination  of  the  RL  and  of  its  variabiHty. 

(i)  Inspection  of  Resiclts :  the  Course  of  the  Two-point  Judg- 
ments.— The  limen  of  dual  impression  upon  the  skin,  like  all  other 
psychophysical  limens,  is  a  variable  magnitude,  subject  to  the 
influence  of  accidental  errors.  And  the  values  found  for  it,  hke 
the  values  found  for  the  other  Hmens,  obey  a  certain  law  of  dis- 
tribution, may  be  arranged  on  a  certain  scheme  of  frequencies. 
The  precise  formulation  of  this  law  depends,  of  course,  upon  the 
special  conditions  of  experiment.  In  general,  however,  it  may  be 
represented  graphically  by  a  curve  of  the  form  shown  in  Fig.  29. 
The  values  of  the  abscissas,  from  o  onwards,  here  stand  for  the 
various  separations  D  of  the  compass-points.  The  curve  is  the 
curve  of  distribution  of  the  values  found  for  the  RL  :  that  is  to 
say,  the  ordinate  drawn  vertically  to  meet  the  curve  from  any 
given  point  upon  the  Hne  of  abscissas  represents  the  probability 
{and  therefore,  in  a  long  series  of  experiments,  the  relative  fre- 


94  The  Metric  Methods 

quency)  of  the  case  that  the  RL  corresponds  to  that  particular 
Z>-value.  Hence  the  area  o  D^  p^  p^  represents  the  probability 
(relative  frequency)  of  the  case  that  the  RL  is  <  Z)^  i.  e.^  that 
the  stimulus  D^  evokes  the  two-point  judgment.  Similarly,  the 
area  o  D^  p^  p^  represents  the  probability  (relative  frequency)  of 


the  case  that  the  RL  is  <  Z>2,  i.  ^.,  that  the  stimulus  D^  evokes 
the  two-point  judgment :  and  so  on.  The  RL  is  not  a  fixed  value ; 
it  is  subject,  in  the  single  observation,  to  the  play  of  all  sorts  of 
accidental  errors. 

The  curve  drawn  in  Fig.  29,  schematic  though  it  is,  enables  us 
to  estimate  at  a  glance  the  value  of  a  series  of  results  obtained 
by  the  method  now  under  discussion.  We  see  that,  as  D  in- 
creases, the  curve  first  rises  to  a  maximal  height,  and  then  grad- 
ually falls  again  to  meet  the  axis  of  abscissas.  This  means  that 
our  percentages  of  two-point  judgments  (p.  93)  should  increase, 
at  first  quickly  and  then  more  slowly,  with  increase  of  D^  until 
finally  a  point  is  reached  at  which  no  other  judgments  are  re- 
corded. A  really  first-rate  set  of  results  will,  as  a  matter  of  fact, 
evince  this  uniformity. 

The  results  which  we  have  quoted  diverge  at  two  points  from 
the  ideal  law  of  distribution.  Notice,  first  of  all,  that  the  30%  of 
two-point  judgments  for  D  =  o  is  altogether  anomalous.  This 
value  of  D  ought  to  have  given  (and  in  good  experimental  series 
does  give)  a  percentage  of  two-point  judgments  that  is  less 
than  the  percentage  found  with  the  lowest  positive  value  of  D. 
In  the  present  instance,  it  gave  more  than  twice  as  many  two- 
point  judgments  as  were  recorded  for  Z>=  i  Paris  line.  A  diver- 
gence of  this  sort,  where  the  percentage  of  two-point  judg- 
ments for  a  given  D  is  the  same  as  (or  greater  than)  the  per- 
centage value  for  the  next  larger  Z>,  is  termed  an  inversion  of  the 
first  order.     We  have  every  reason  to  suppose,  from  the  figures. 


§   20.   The  Method  of  Constant   Stimuli  95 

that  the  experimental  procedure  with  D  ==o  varied  in  some  way 
from  the  procedure  followed  with  the  other,  positive  /^-values : 
perhaps  the  pressure  of  the  single  point  was  stronger  than  that 
of  the  two  points  ;  perhaps  some  misleading  suggestion  was  given 
to  O.  We  cannot  say  what  the  error  was,  but  we  may  be  sure 
that  an  error  was  involved ;  and  we  must  accordingly  leave  the 
percentage  for  Z?  =  o  out  of  account  in  our  later  calculations. 

Notice,  secondly,  that  the  difference  between  0.87  and  0.96 
{B  =  4  and  5  Paris  lines)  is  not  less,  but  somewhat  greater,  than 
the  difference  between  0.80  and  0.87  (1^=3  and  4).  The  dif- 
ferences of  the  successive  percentage  values,  between  the  limits 
Z?  =  o.5  and  Z>=6,  are  0.04,  0.26,  0.25,  0.15,  0.07,  0.09,  0.04. 
With  the  one  exception  of  the  0.09  following  the  0.07,  the  se- 
quence is  perfectly  regular.  A  divergence  of  this  sort — where, 
as  D  increases,  the  percentage  of  the  two-point  judgments  in- 
creases at  first  with  diminishing  and  then  with  increasing  rapidity, 
— is  termed  an  inversion  of  the  second  order.  The  inversion  is 
here  so  slight  that  we  need  not  reject  the  results  affected  by  it. 
At  the  same  time  its  presence  is  an  indication  of  some  fault  of 
procedure,  on  the  part  of  E  or  of  O. 

Expressed  in  terms  of  Fig.  29,  an  inversion  of  the  second  order 
evidently  means  that  the  curve  of  distribution,  instead  of  run- 
ning smoothly  and  evenly,  drops  down,  at  some  point  of  its  course, 
towards  the  axis  of  abscissas,  and  then  rises  again  to  resume  its 
normal  progress.  An  inversion  of  the  first  order  means  that  the 
curve  falls  at  some  point  as  low  as  the  axis  of  abscissas  (or 
even  below  that  axis) ,  and  then  rises  again  to  run  its  course 
above  it. 

(2)  Calculation  of  the  RL:  First  Procedure. — The  RL  corre- 
sponds, by  definition,  to  that  value  of  Z?  which  calls  forth  50%  of 
two-point  judgments,  and  50%  of  one-point  or  doubtful  judg- 
ments. Our  results  show  that  /?=  1.5  gives  40%,  while  D=2 
gives  65  %,  of  two-point  judgments.  Plainly,  then,  the  liminal  D 
lies  somewhere  between  the  limits  1.5  and  2.0  Paris  lines.  Let 
us  represent  by  D^  the  D  that  gives  the  least  relative  number 
of  two-point  judgments  >  0.5  ;  by  A  the  D  that  gives  the  great- 
est relative  number  <  0.5  ;  and  by  n^  and  n^  the  corresponding 
relative  numbers  of  the  two-point  judgments  themselves.     We 


96  The  Metric  Methods 

thus  obtain  the  formula  (already  employed  in  a  different  con- 
nection on  p.  86) : 

Working  this  out,  we  find  : 

RL^  1.7  Paris  lines. 
The  formula  is  not  altogether  adequate,  and  not  entirely  free 
from  arbitrariness  (see  Question  5,  p.  91);  so  that  the  result 
must  be  considered  merely  approximative. 

If  we  follow  this  procedure  for  the  determination  of  the  RL^ 
we  have  unfortunately  no  means  of  measuring  its  variability  ; 
the  formula  does  not  yield  a  measure  of  precision.  We  may, 
however,  attempt  roughly  to  determine  whether  the  curve  of 
distribution  of  the  accidental  values  of  the  RL  is  symmetri- 
cal or  asymmetrical.  Thus  (a)  we  may  pick  out  two  Z^'s  that 
lie  at  about  the  same  distance  above  and  below  the  calculated 
RL^  and  see  whether  the  corresponding  /^-values  differ  by  the 
same  or  by  different  amounts  from  ^^  =  0.5.  If  the  differences 
are  approximately  the  same,  the  curve  of  distribution  is  approxi- 
mately symmetrical  between  the  Z^-limits  employed  ;  if  they  are 
clearly  not  the  same,  then  the  curve  is  asymmetrical.  Thus : 
for  D  =  i.5=i?Z  — 0.2  and  2.o  =  i?Z+o.3, 
0.5— ;^     =  +0.10  and  —0.15. 

Or  again : 

for      D    =     o. S=RL— 1.2     and     3.o  =  i?Z+i.3, 
0.5 — n     =  -fo.40  and  — 0.30. 

The  first  pair  of  results  might  stand  for  symmetry,  since  the 
+  0.3  is  slightly  greater  than  the  — 0.2.  The  second  term  of 
the  second  pair,  however,  is  not  only  not  greater, — it  is  actually 
smaller  than  the  first  term.  This  means  that,  in  the  region  of 
the  B's  that  are  >  RL,  the  curve  drops  toward  the  axis  of 
abscissas  more  quickly  than  it  rises  from  that  axis  in  the  region 
of  the  smaller  D's.  {b)  The  same  result  comes  out  if  we  select 
two  /^-values  that  differ  by  approximately  the  same  amount,  plus 
and  minus y  from  0.5,  and  compare  the  differences  between  the 
corresponding  Z)-values  and  the  value  of  the  RL.  Thus  : 
for  0.5— ;/  =  +0.40  and  —0.37, 
RL—D         =      +1.2       and     —2.3. 


§  20.   The  Method  of  Constant  Stimuli  97 

The  difference  between  RL  and  D  in  the  upper  region  of  the 
scale  is  greater  (for  approximately  the  same  difference  between 
0.5  and  n)  than  it  is  in  the  lower :  /.  e.^  again,  the  curve  drops 
toward  the  axis  of  abscissas  more  quickly  in  the  region  of  the 
large  Z)'s  than  it  rises  from  that  axis  in  the  region  of  the 
small  Z^'s.  Had  the  {RL — Z>)-values  been  approximately  the 
same,  we  should,  on  the  contrary,  have  inferred  that  the  curve, 
between  the  limits  D  =  o.^  and  Z>  =  4.0,  ran  an  approximately 
symmetrical  course. 

Further  than  this,  towards  a  knowledge  of  the  variability  of 
the  RL^  the  present  procedure  cannot  take  us. 

Result  {b)  is  shown  still  more  clearly  if  we  take  the  two  Z>'s  i.o  and  4.0, 
We  then  have : 

for  0.5  — «     =      +0.36     and     —0.37, 

RL  —  D  =      +0.7       and     —2.3. 

Since,  however,  the  value  0.87  for  D  =  4.o  is '  probably  too  small  (inversion 

of  the  second  order),  it  is  better  to  compare  it,  as  we  have  done,  with  the 

value  o.io  forZ>  =  o.5. 

(3)  Calculation  of  the  RL  :  Second  Procedure. — We  have  said 
that  the  RL  is  a  variable  magnitude.  Let  us  now  represent  by  the 
same  symbol,  RL^  the  median  value  of  the  limen  of  dual  im- 
pression, and  by  ±  S  the  magnitude  of  the  accidental  variation  to 
which  this  value  is  subject;  so  that  thfe  symbol  (i?Z±8)  will 
stand  for  the  limen  of  dual  impression  as  affected  by  accidental 
errors.  A  two-point  judgment  will  then  be  given  by  O  in  all 
cases  in  which  />  is  >  {RL  db  8).  If  we  make  a  long  series  of 
experiments  with  the  same  D  and  under  the  same  conditions,  we 
may  say  that  the  n  corresponding  to  this  D  indicates  the  proba- 
bility that  Z>  is  >  {RL±h). 

(a)  Suppose,  now,  that  Z)  is  >  RL.  Then  D  must  be 
>(i?Z±S),  first  whenever  §  is  negative,  and  secondly  when  it  is 
positive  but  in  absolute  magnitude  <i{D~RL).  Since  RL  is 
a  median  value,  the  probability  that  8  is  negative  is  =  0.5.  The 
probability  that  it  is  positive  but   <C(L>—RL)  is  given  by  the 

/+(Z>-i?Z) 
fi±S)dS,  where  the  term/(±S)   is  a  general  ex- 
o 

pression  for  the  probability  of  the  occurrence  of  the  magnitude 
G 


98  The  Metric  Methods 

±8.  Hence  the  probability  that  Z>  is  >(i?Z±S)  is  the  sum  of 
these  two  probabilities  :  or,  in  other  words,  if  D  is  yRL  and  n 
consequently  >0.5,  the  equation  holds,  for  a  long  series  of  ex- 
periments, that 

«=!-+ j  /(±8)^S. 

{b)  Suppose,  again,  that  D  is  <^RL.  Then  D  will  be  > 
(T^ZiS)  only  when  8  is  negative  and  in  absolute  magnitude 
y{RL—D).  The  probability  of  this  case  is  =  the  probability 
(0.5)  that  8  is  negative,  minus  the  probability  that  it  is  negative 
and  lies  between  the  limits  o  and  —  {RL  —  D).  In  other 
words,  if  D  is  <^  RL  and  n  consequently  <o.5,  the  equation 
holds,  for  a  long  series  of  experiments,  that 


t-  r/(±8)^8. 

^    —(R 


1 

{RL-D) 

(c)  Lastly,  if  ^2  =  0.5,  D  must  be  regarded  3is  =  RL. 


So  far,  our  equations  contain  only  the  indeterminate  expression 
y*(±8).  Let  us  now  rewrite  them,  in  terms  of  Gauss'  law  of 
error.     We  then  obtain,  as  a  general  formula, 

(U-RL) 


or,  if  we  make  ^8  =  ^, 

where  the  difference  (D—RL)  is  positive,  o  or  negative,  accord- 
ing as  ;2  is>,  =  or  <  0.5. 

This  equation  contains  precisely  the  two  values  which  we  are 
seeking  to  determine  :  RL,  the  representative  value  of  the  limen 
of  dual  impression,  and  /^,  the  measure  of  its  precision.  How 
are  we  to  solve  it  ? 

Let  us  turn  back  to  our  experimental  results.  We  throw  out 
the  series  for  D  =  o,  because  the  ^2  =  0.30  is  anomalous  (inver- 
sion of  the  first  order).  We  throw  out  also  the  series  for  D  =  6, 
because  the  n  =  i  shows  that  we  have  passed  beyond  the  limits 
of  variability  of  the  RL.     We  are  left  with  the  seven  Z?'s  0.5  to 


§  20.    The  Method  of  Constant  Stimuli  99 

5.0,  and  with  the  corresponding  /^'s.  Call  the  Z^'s  Z>i,  D^y 
/?3,  .  .  .  Z>7,  and  the  //'s  ;2i,  ^/gj  ?^3,  •  •  •  '^h-  Our  equation  en- 
ables us  to  derive  from  n^  a  definite  numerical  value  /i  of  the 
product  (^Di—RL)h,  from  n^  a  similar  value  t^  of  the  product 
{D^—RLyiy  and  so  on.  The  following  Table  gives  the  values 
of  t  for  determinate  values  of  n. 


FECHNER  S    FUNDAMENTAL    TABLE 


n 

t  =  hZ 

n 

/=//8 

n 

t=hl 

0.50 

0.0000 

0.67 

0.3111 

0.84 

0.7032 

0.51 

0.0177 

0.68 

0.3307 

0.85 

0.7329 

0.52 

0.0355 

0.69 

0.3506 

0.86 

0.7639 

0.53 

0.0532 

0.70 

0.3708 

0.87 

0.7965 

0.54 

0.0710 

0.71 

0.3913 

0.88 

0.8308 

0.55 

0.0890 

0.72 

0.4121 

0.89 

0.8673 

0.56 

0.1068 

0.73 

0.4333 

0.90 

0.9062 

0.57 

0.1247 

0.74 

0.4549 

0.91 

0.9481 

0.58 

0.1428 

0.75 

0.4769 

0.92 

0.9936 

0.59 

0.1609 

0.76 

0.4994 

0.93 

1.0436 

0.60 

0.1 791 

0.77 

0.5224 

0.94 

1.0994 

0.61 

0.1975 

0.78 

0.5460 

0.95 

1.1631 

0.62 

0.2160 

0.79 

0.5702 

0.96 

1.2379 

0.63 

0.2347 

0.80 

0.5951 

0.97 

1.3297 

0.64 

0.2535 

0.81 

0.6208 

0.98 

1.4522 

0.65 

0.2725 

0.82 

0.6743 

0.99 

1.6450 

0.66 

0.2917 

0.83 

0.6747 

1. 00 

00 

If  «  is  <  0.5,  look  in  the  Table  not  for  n  but  for  \—n^  and  take  /  nega- 
tive.    Thus  the  /  for  «  =  0.25  is  — 0.4769. 


100  The  Metric  Methods 

By  help  of  this  Table  we  may  write  the  equations 

t^  =  {D^—RL)h, 

t,  =  {D^—RL)h 
in  numerical  form,  as  follows  : 

—0.9062  =  {o.$—RL)^, 
—o.y6s9  =  (i.o—RL)k, 
— o.  1 79 1  =  ( 1 . 5  —RL)h, 
0.272$  =  {2.0— RL)h, 
o.595i  =  (3.o— y?Z)/^, 
0.7965  =  (4.0— i^Z)//, 
1.2^79=  {S.o—RL)h. 

These  equations  can  be  solved,  for  RL  and  h,  by  the  Method  of 
Least  Squares. 

So  far,  so  good !  We  are  not  yet,  however,  out  of  the  mathe- 
matical wood.  If  we  were  to  solve  the  equations  as  they  stand, 
we  should  be  making  a  mistake  in  theory,  and  a  mistake  which 
is  by  no  means  always  negligible  in  practice.  We  should  be  pro- 
ceding  as  if  A,  tif  etc.,  were  observed  values.  Now  the  values 
really  observed  are  not  these  /-values,  but  the  //-values.  We 
must,  therefore,  seek  (so  to  say)  to  transform  the  /-values  into 
observed  values  ;  and  we  may  do  this  by  compensating  the  error 
which  their  direct  treatment  as  observed  values  would  involve. 
We  may  do  it,  in  other  words,  by  weighting  the  /-values. 

Each  of  the  values  ni,  «2>  etc.,  has  a  weight  w'  proportional  to 
the  number  of  observations  upon  which  it  is  based.  This  value 
w'  must  for  our  purposes  be  multiplied  by  a  coefficient  w'\  to 
be  determined  from  the  following  Table.  Then  the  products 
w\  w'\,  w\  w'^y  etc.,  are  the  required  weights  of  the  values 
'i>  ^2>  etc. 


§•  20.   The  Method  of  Constant  Stimuli  lor 


MULLERS 

TABLE    OF    ( 

COEFFICIENTS 

OF    WEIGHTS 

n 

w" 

n 

w" 

n 

w" 

0.50 

1. 000 

0.67 

0.824 

0.84 

0.373 

0.51 

0.999 

0.68 

0.803 

•       0.85 

0.342 

0.52 

0.997 

0.69 

0.782 

0.86 

0.31 1 

0.53 

0.994 

0.70 

0.760 

0.87 

0.281 

0.54 

0.990 

0.71 

0.737 

0.88 

0.251 

0.55 

0.984 

0.72 

0.712 

0.89 

0.222 

0.56 

0.977 

0.73 

0.687 

0.90 

0.193 

0.57 

0.969 

0.74 

0.661 

0.91 

0.166 

0.58 

0.960 

0.75 

0.634 

0.92 

0.139 

0.59 

0.950 

0.76 

0.606 

0.93 

0.114 

0.60 

0.938 

0.77 

0.578 

0.94 

0.089 

0.61 

0.925 

0.78 

0.550 

0.95 

0.067 

0.62 

0.91  I 

0.79 

0.521 

0.96 

0.047 

0.63 

0.896 

0.80 

0.492 

0.97 

0.029 

0.64 

0.880 

0.81 

0.463 

0.98 

0.014 

0.65 

0.862 

0.82 

0.433 

0.99 

0.004 

,0.66 

0.843 

0.83 

0.403 

The  weight  of  an  n  which  is  <C  0.5  is  the  same  as  the  weight  of  an  n 
which  exceeds  0.5  by  the  same  amount.  Thus  the  weights  of  n  =  o.2^ 
and  of  «  =  o.75  are  both  alike  =  0.634. 


102  The  Metric  Methods 

Since,  as  a  general  rule,  the  values  of  n  are  determined  from 
the  same  number  of  observations,  the  weights  w'  will  in  most 
cases  all  be  =  i,  and  the  factor  w'  in  the  product  w'  w"  may  ac- 
cordingly be  neglected.  Let  us  term  this  product  w.  Our  equa- 
tions now  become : 

w^ti  =  {D^—RL)k, 

w^t^  =  {D2—RL)h, 

w,tT  =  {D^—RL)h. 
Or,  in  numerical  form  (on  the  assumption  that  w^  =1), 
—0.9062  (0.193)  =  (0. 5— i?Z)//, 
—0.7639  (o.3ii)  =  (i.o— 7?^)//, 
-0.1791  (o.938)  =  (i.5-i?Z)/^, 
0.2725   {0.^62)  =  {2.0— RL)h, 
0.5951   (0.492)  =  (3.o-ieZ)//, 
0.7965   (0.28 1)  =  (4.0— i?Z)//, 
1.2379  (o.047)  =  (5.o— 7eZ)>^. 
Treating   these  equations  by  the  Method  of  Least  Squares,  we 
obtain  the  normal  equations  : 

[D^w]h  —  [Dw']RL.h  =   [Dtw], 
—  [Dwyt  4-  [    w\RL.h  =  —[tw\. 
The  squared  bracket  is  here  used  as  the  sign  of  summation  :  so  that 
[D^w']  means  {D^^Wi  ■\-  D^^.'^^-^-   •  .  •  ^^^7);    [T>w]   means 
(DiWi  +  D2'W2+    .  .  .  I>7Wt);  and  so   on.      A  rough  solution, 
with  rounding  of  the  fractions  to  two  decimal  places,  gives  : 
i6.05>^  —  6.377?^.// =  1.96, 
—  6.37^  -I-  i,\2RL.h  =  0.23; 

whence  we  find 

h  =  0.49, 
RL  =  1.88. 

To  those  who  are  not  mathematically  inclined,  the  last  few  pages  may 
have  appeared  somewhat  bewilderingly  difficult.  Let  it  be  said,  then,  that 
the  mathetnatical  operations  involved  are  exceedingly  simple.  They  consist 
(i)  in  determining  from  Fechner's  Table  the  /-values  for  the  given  ?^-values ; 
(2)  in  determining  from  Miiller's  Table  the  ^''-values  for  the  same  ^-values ; 
and  (3),  after  the-full  series  of  these  determinations  has  been  made,  in  ap- 
plying the  scheme  of  the  Method  of  Least  Squares,  as  indicated  above,  to 
a  half-dozen  equations  that  contain  two  unknown  quantities.  So  far  as  the 
calculations  go,  it  is  all  a  matter  of  simple  arithmetic.     The  U7tder standing 


§  21.   The  Method  of  Constant  Stimuli  103 

of  the  mathematical  argument  is  another  thing.  Its  general  course  ought, 
however,  to  be  intelligible  from  §  15;  and  the  derivation  of  the  normal 
equations  is  explained  in  detail  in  the  mathematical  text-books.  Even 
without  an  adequate  understanding  of  the  argument,  the  reader  should  be 
able  to  realise  the  superiority  of  this  second  over  the  first  procedure  (p.  95). 
We  have  determined  RL^  as  we  desired ;  we  have  also  determined  ^, 
the  measure  of  precision  (p.  44  above);  and  in  so  doing,  we  have  utilised 
not  two  selected  «,  but  the  whole  number  of  «  for  the  different  Z>'s  at  our 
disposal,  allowing  each  its  proper  weight  in  the  final  result.  The  game  has 
been  well  worth  the  candle. 

EXPERIMENT  XX 

Determination  of  the  Limen  of  Dual  Impression  upon  the  Skin, 
— Materials. — iEsthesiometric  compasses.     Mm.  paper  scale. 

Directions. — Having  chosen  the  part  of  the  skin  upon  which 
the  RL  is  to  be  determined,  E  ascertains,  in  preliminary  experi- 
ments, the  value  of  the  upper  D.  He  then  makes  out  a  series  of  R^ 
which  should  consist  of  about  10  terms,  between  the  Hmits  o 
and  this  upper  D.  The  10  ^-values  are  written  upon  10  card- 
board tickets;  these  are  shuffled,  or  shaken  in  a  bag,  and  the 
order  of  their  drawing  gives  the  order  of  the  first  series  of  appli- 
cations. 

Each  ^- value  is  to  be  used  50  times  over,  so  that  the  whole 
number  of  applications  is  500.  E  may  therefore  draught  a  plan 
of  the  complete  experiment  beforehand.  He  first  makes  out  25 
series,  drawing  at  haphazard  from  the  bunch  of  tickets ;  then 
inverts  these  series  (writes  them  backwards)  ;  and  then  shuffles- 
the  50  papers,  and  works  through  the  series  in  the  resulting 
haphazard  order. 

Care  must  be  taken  not  to  fatigue  the  skin  ;  and  warming-up 
experiments  must  be  given  at  the  beginning  of  every  laboratory 
session. 

Questions. — E  and  O  (\)  Why  has  this  Method  of  Constant 
StimuU  received  the  name  of  the  method  of  right  and  wrong 
cases } 

(2)  In  determining  RL  and  h  by  our  second  procedure,  we 
have,  after  all,  used  only  a  part  of  our  results  :  viz.,  the  two-point 
judgments.  We  have  done  nothing  with  the  one-point  and 
doubtful  judgments.  Can  these  outstanding  judgments  be  put  to 
any  psychophysical  purpose } 


104  ^^^^  Metric  Methods 

(3)  At  a  certain  stage  of  the  argument,  we  assumed  that  the 
quantities  with  which  we  are  dealing  were  distributed  in  accord- 
ance with  Gauss'  Law.     How  can  we  test  this  assumption  ? 

(4)  In  deriving  our  formulae  for  n  on  pp.  97  f.  we  spoke  of  the 
limen  as  a  variable  magnitude,  ranging  under  the  influence  of 
accidental  errors  between  the  limits  RL—8  and  RL-\-S',  and  we 
regarded  D  as  constant,  i.  ^.,  as  unaffected  by  the  accidental  errors. 
We  should  arrive  at  just  the  same  result  if  we  took  D  as  the  va- 
riable and  RL  as  the  constant  magnitude.  Which  is  the  preferable 
point  of  departure  for  the  argument,  and  why  .? 

(5)  We  find  a  certain  small  percentage  of  two-point  judgments 
when  D  is  =  o.  What  is  the  mathematical  expression  of  this 
fact  ? 

(6)  We  have  spoken  in  the  text  only  of  the  accidental  errors 
±8,  whose  power  to  prompt  an  one-point  judgment  decreases  as 
D  increases.  May  we  suspect  the  presence  of  any  other  kind  of 
accidental  errors  ? 

(7)  Suggest  further  experiments  by  the  Method. 

§  21.  The  Determination  of  Equivalent  R  by  the  Method  of 
Constant  Stimuli. — We  said  above  (§17)  that  an  experiment 
would  presently  be  assigned  in  which  the  Method  of  Average 
Error  should  be  employed  in  a  modified  form.  The  problem 
of  that  method,  it  will  be  remembered,  is  the  equation  of  two  R, 
A  constant  stimulus,  r,  is  presented  by  Ey  and  O  is  required  to 
make  another  stimulus,  ri,  subjectively  equal  to  it. 

We  now  undertake  the  equation  of  two  R  by  help  of  the 
Method  of  Constant  Stimuli.  We  thus  avoid  all  the  complica- 
tions that  arise  from  (9's  active  manipulation  of  ri. 

Materials. — Miinsterberg's  apparatus  for  the  comparison  of 
arm  movements.  [The  apparatus.  Fig.  30,  consists  essentially 
of  a  car  which  travels  smoothly  on  three  horizontal  tracks.  The 
car  carries  a  hollow  cylinder  of  brass  for  the  reception  of  (9's 
fore-finger,  and  a  pointer  which  traverses  a  scale  marked  on 
one  of  the  outer  tracks.  Two  sliding  blocks,  which  can  be  set 
at  any  point  along  the  middle  track,  mark  the  beginning  and  end 
of  the  two  movements.  The  stand  which  supports  the  tracks  is 
adjustable  in  height. —  Cords  run  from  the  two  ends  of  the  car^ 


§   21.    The  Determination  of  Equivalent  R 


105 


over  pulleys,  to  weight  holders ;  and  the  iron  standard  has  a  set- 
screw  and  slitted  arc  by  means  of  which  the  tracks  may  be  set 


Fig.  30. 

obliquely  or   even  vertically.     These  parts  of  the  apparatus  do 
not  now  concern  us.  ] 

Preliminaries. — E  adjusts  the  height  of  the  apparatus  so 
that  Oy  standing  squarely  to  it,  can  move  the  car  without  effort. 
The  height  of  the  tracks  above  the  floor  is  measured,  and  the 
position  of  (9's  feet  and  of  the  feet  of  the  standard  is  indicated 
by  chalk  marks :  it  is  necessary  that  these  determinations  re- 
main constant  from  day  to  day.  The  standard  distance  ( say,  of 
40  cm. )  is  marked  off  by  the  sliders ;  and  the  standard  stimulus 
consists  always  in  the  movement  of  the  car  by  O^  from  left  to 
right,  between  these  fixed  limits. 


EXPERniENT    XXI 

Can  you  now,  from  what  you  know  of  the  Method  of  Average 
Error  and  the  Method  of  Constant  Stimuli,  work  out  a  method 
for  the  determination  of  the  subjective  equivalent  of  this  stand- 
ard r  of  40  cm.- movement }  Remember  that  no  adjustments 
are  to  be  made  by  O,     Remember  also  to  guard  against  constant 


io6  The  Metric  Methods 

errors.  Make  out  a  plan  of  the  experiment,  as  well  as  you  can, 
.and  submit  it  to  the  Instructor. 

Questions.  —  E  and  (9  (i)  Have  you  any  criticism  to  offer 
upon  the  apparatus } 

(2)  Is  judgment  passed  purely  in  terms  of  kinaesthetic  sen- 
sations ?     If  not,  what  other  factors  enter  into  it  ? 

§  22.  The  Method  of  Constant  Stimulus  Differences  (Method  of 
Eight  and  "Wrong  Cases). — The  object  of  this  Method  is,  to  deter- 
mine the  DL  and  its  measure  of  precision,  as  the  object 
of  the  Method  of  Constant  StimuH  is  to  determine  the  RL  and 
its  h.  The  procedure  in  the  two  cases  is  strictly  analogous.  We 
may  illustrate  the  course  of  the  present  method  by  reference  to 
lifted  weights. 

We  take  as  our  standard,  5,,  a  weight  of,  say,  107 1  gr.  This 
is  the  weight  whose  upper  and  lower  DL  are  to  be  found.  We 
provide  ourselves  with  a  graduated  series  of  weights  of  compar- 
ison, ^.-  these  may,  ^.  ^.,  be  weights  of  921,  971,  1021,  1071, 
1121,  1171,  1221  gr.  One  of  these  is  identical  with  the  stand- 
ard ;  two  differ  from  it  by  ±  50,  two  by  ±  100,  and  two  by  ± 
150  gr.  The  standard  weight  is  to  be  compared  with  each  of 
the  weights  of  comparison,  taken  in  haphazard  order,  50  times 
over,  and  the  judgments  are  to  be  recorded  and  classified.  O 
judges  always  in  terms  of  the  weight  last  Hfted,  or  lifted  second ; 
and  his  judgments  may  take  the  form  <  much  greater ,'  <  greater,  * 
*  doubtful,'  'less,'  'much  less.'  That  is  to  say,  he  judges  of 
the  second  weight  lifted  in  a  comparative  observation  as  *  much 
greater, '  '  less, '  etc.,  than  the  weight  first  lifted. 

The  judgments  are  to  be  abbreviated  on  the  record  sheet  as  »,  >,  ?, 
<,  «.  If  by  chance  the  two  weights  appear  to  O  to  be  positively  equal, 
the  judgment  =  is  to  be  entered  in  the  record,  but  the  equal-judgments  are 
to  be  counted  with  the  doubtfuls  for  purposes  of  calculation. 

Now,  however,  that  we  are  working  with  the  two  Ry  there  are 
two  possible  time  orders  :  the  weight  first  hfted  may  be  either 
5  or  one  of  the  Cs.  If  we  are  to  control  the  constant  error  of 
time,  the  experiment  must  be  performed  twice  over  :  once  with 
5  lifted  first,  and  again  with  C  hfted  first.     Our  50  comparisons 


§  22   The  Method  of  Constant  Stimulus  Differences    107 

are  thus  increased,  for  each  (7,  to  100 ;  we  must  make  in  all  700 
(not  350)  comparisons. — We  need  not  consider  here  the  con- 
stant error  of  space,  as  it  is  ruled  out  by  a  special  arrangement 
of  the  apparatus  (see  p.   116  below). 

The  results  of  a  complete  experiment  will  resemble  those 
shown  in  the  following  Table.  The  five  classes  of  judgments 
are  here,  for  simplicity's  sake,  reduced  to  three  ;  and  the  judg- 
ments given  in  the  first  time  order  (5  first)  have  been  reversed, 
in  order  that  all  judgments  alike  may  refer  to  5.  Thus  a 
< -judgment  in  the  Table  means  that  5  was  judged  as  lighter 
than  the  C  oi  the  same  horizontal  line;  a  > -judgment  means 
that  5  was  judged  as  heavier  than  the  C. 

Standard  weight  =1071  gr. 


Weight  of 

Time  order  I. 

Time  order  II. 

Comparison 

(5  first) 

{S 

second) 

< 

? 

> 

< 

?           > 

921 

0 

2 

48 

I 

6         43 

971 

0 

7 

43 

6 

4         40 

1021 

3 

18 

29 

7 

19         24 

1071 

13 

18 

19 

17 

19         14 

1121 

27 

16 

7 

28 

18           4 

1171 

36 

II 

3 

40 

9           I 

1221 

47 

2 

I 

43 

7          0 

(i)  Inspection  of  Results:  the  Course  of  the  Judgments. — The 
upper  and  lower  DL  are  variable  magnitudes,  and  the  values 
found  for  them  obey  a  certain  law  of  distribution,  may  be  ar- 
ranged on  a  certain  scheme  of  frequencies.     This  law  may  be 


represented  graphically,  for  the  upper  DL,  by  a  curve  of  the  form 
shown  in  Fig.  31.  The  values  of  the  abscissas,  from  o  onwards, 
here  stand  for  the  various  weights  :  the  line  os  represents  the 
standard  weight  or  5.     The  curve  is,  as  we  have  said,  the  curve 


io8  The  Metric  Methods 

of  distribution  of  the  values  found  for  the  upper  DL :  that  is 
to  say,  the  ordinate  drawn  vertically  to  meet  the  curve  from  any 
given  point  c^  or  ^2  upon  the  line  of  abscissas  represents  the 
probability  (and  therefore,  in  a  long  series  of  experiments,  the 
relative  frequency)  of  the  case  that  the  DL  corresponds  to  the 
+  - difference  sc^  or  sc^,  between  S  and  the  weight  of  com- 
parison C.  Hence  the  area  ac^  p\  represents  the  probability 
(relative  frequency)  of  the  case  that  a  C^oc^  is  judged  >  5; 
and  the  area  ac^  p^  represents  the  probability  (relative  fre- 
quency) of  the  case  that  a  C=0Ci  is  judged  >  5.  A  similar 
construction  may  be  made  for  the  lower  DL. 

We  see  that,  as  C  increases,  the  curve  first  rises  to  a  maximal 
height,  and  then  gradually  falls  again  to  meet  the  axis  of  abscis- 
sas. This  means  that  our  percentages  /  of  <  less '  judgments 
should  increase,  at  first  quickly  and  then  more  slowly,  with  in- 
crease of  Cy  until  finally  a  point  is  reached  at  which  no  other 
judgments  are  recorded.  The  figure  for  the  lower  DL  would 
show,  similarly,  that  the  percentages  ^  of  *  greater '  judgments 
should  decrease,  at  first  quickly  and  then  more  slowly,  with  in- 
crease of  Cy  until  finally  no  such  judgments  are  recorded.  If 
we  find  that  our  results  show,  for  some  increase  of  C,  no  increase 
of  /  or  decrease  of  gy — while  the  ^''s  themselves  are  neither  so 
small  that  /=o  and  g=iy  nor  so  large  that  g=o  and  /=  i, — 
we  have  an  inversion  of  the  first  order.  There  is  no  inversion 
of  this  kind  in  the  results  quoted  on  p.  107.  If  we  find,  under 
the  same  conditions,  that  /  increases,  or  that  g  decreases,  at  first 
with  decreasing  and  then  with  increasing  rapidity,  we  have  an 
inversion  of  the  second  order.  All  four  columns  of  the  Table 
contain  an  inversion  of  this  second  sort. 

Let  us,  first  of  all,  rewrite  the   Table 
(less),  u  (uncertain),  g  (greater)  replace  the  original  <,  ?,  > 

C 

921 

971 
1021 
1071 
1121 
1171 
1221 


/ 

u 

g 

0.00 

0.04 

0.96 

0.00 

0.14 

0.86 

0.06 

0.36 

0.58 

0.26 

0.36 

0.38 

0.54 

0.32 

0.14 

0.72 

0.22 

0.06 

0.94 

0.04 

0.02 

ntages. 

The  s 

ymbols  / 

<  ?,  >. 

We  obtain  : 

/ 

u 

g 

0.02 

0.12 

0.86 

0.12 

0.08 

0.80 

0.14 

0.38 

0.48 

0.34 

0.38 

0.28 

0.56 

0.36 

0.08 

0.80 

0.18 

0.02 

0.86 

0.14 

0.00 

§  22.   The  Method  of  Constant  Stimulus  Differences    109 

(I)  There  is,  we  said,  no  inversion  of  the  first  order.  It  is  true  that  we 
have 7=  0.00  in  two  cases,  in  neither  of  which  ^  is  =1.  This  strict  paral- 
lelism of  the  course  of  /  and  g  is,  however,  not  essential.  The  results  are 
regular  when  they  conform  to  any  one  of  the  three  following  schemata : 


c 

I 

u 

g 

/ 

u 

g 

/ 

u 

g 

x^y 

0.00 

0.00 

I. GO 

0.00 

0.00 

1. 00 

0.00 

0.0 1 

0.99 

X 

0.00 

0.00 

1. 00 

0.00 

0.0 1 

0.99 

0.00 

0.02 

0.98 

.00 

.06 

.20 

.28 

.18 

.22, 

.10 

.02 

.20 

.22 

.24 

.06; 

.10 

.28 

.20 

.24 

.08 

.04, 

.06 

.32 

.20 

.20 

.06 

.02. 

the  third  of  which  is  represented  in  the  Table. — The  same  thing  holds, 
mutatis  mutandis,  for  high  values  of  C  that  give  g=  0.00.  (2)  The  inver- 
sions of  the  second  order  become  clear  as  soon  as  we  write  out  the  differ- 
ences between  the  successive  terms  of  the  four  relevant  columns.  These 
differences  are  : 

/ 

g 

The  irregularity  of  the  last  series  is,  perhaps,  too  slight  to  be  properly 
termed  an  inversion.  The  curve  rises  steeply,  drops,  and  then  runs  parallel 
to  the  axis  of  abscissas  before  falling  again  in  the  regular  manner. 

There  is  a  further  point  which  calls  for  notice.  In  a  perfectly- 
regular  set  of  results,  the  ?- judgments  outlast  the  < -judg- 
ments at  the  head,  and  the  > -judgments  at  the  bottom  of  the 
columns.  In  this  regard,  therefore,  the  Table  on  p.  107  conforms 
to  rule.  In  two  of  the  four  cases,  the  <  and  >  judgments  have 
entirely  disappeared,  while  the  }  still  have  the  values  7  and  7  ; 
in  the  other  two,  the  .^-judgments  have  the  numerical  advan- 
tage. 

(2)  Calculation  of  the  DL:  First  Procedure. — The  upper  DL 
corresponds  to  that  value  of  D=^C — 5  which  yields  /^o.5  ; 
the  lower  DL  to  that  value  oi  D^S  —  C  which  yields  ^  =  0.5. 
Availing  ourselves  of  the  formula  given  on  pp.  S6,  96,  we 
find,  approximately  : 

DL„        DL, 

Time  order  1 43  30 

Time  order  II 36  53. 

On  the  basis  of  these  values,  we  may  attempt  roughly  to  deter- 
mine whether  the  curves  of  distribution  of  the  accidental  values 
of  the  DL  are  symmetrical  or  asymmetrical.  To  this  end,  we 
pick  out  two  Z?- values  that  lie  at  about  the  same  distance  above 
and  below  the  calculated  DLy  and  see  whether  the  correspond- 


no  The  Metric  Methods 

ing  /  and  g  values  differ  by  the  same  amount  from  /  and  g 
=  0.50.     Thus: 

Time  order  I.     The  C  that  gives  0.5  /=  1 1 14. 

Fori?  =1114 —  1021^^    93,  0.5  —    /  =.44; 

"               1221  —  1114  =  107,  /    — 0.5  =  .44. 

.The  ^'that  gives  0.5  g  =  104 1. 

For  Z>  =  1041  —    921  =  120,  g  —  0.5  =  .46; 

1171  —  1041  =  130,  0.5  —  ^  ==.44. 

For  I>  =  1041  —    971  =70,  g  —  0.5  =  .36; 

"               1 121  —  1041  =    80,  0.5  —  g  =  .36. 

Time  order  II.     The  C  that  gives  0.5  /=  1 107. 

For  Z>^  1 107 —    971=136,      0.5  —    /    =.38; 

**  1221  —  1 107  =  1 14,        /    —  0.5  =  .36. 

For  D  =  1 107  —  102 1  =86,        g   —  0.5  =  .36; 

"  1 171  —  1107=    64,      0.5  —  g    =  .30. 

The  6" that  gives  0.5  g=  1018. 

For  Z>  =  1018 —    921  =97,       g  —  0.5  =  .36; 

"  1121  —  1018  =  103,     0.5 —    g    =.42. 

What  conclusions  are  to  be  drawn  from  these  figures  ?    . 

(3)  Calculation  of  the  DL  :  Second  Procedure. — A  train  of 
reasoning,  precisely  analogous  to  that  of  pp.  97  f.,  leads  us  to 
the  three  formulae : 


e—^^dtf 


^-^J    iA.n-7 


-r^dt, 


where  the  sign  of  D  is  positive  or  negative  according  as  C  is 
greater  or  less  than  S.  The  equations  contain  the  four  values 
which  we  are  seeking  to  determine  :  DL^  and  DLiy  the  repre- 
sentative values  of  the  upper  and  lower  difference  limens,  and 
h^  and  hy,  their  respective  measures  of  precision. 


§  22.    The  Method  of  Constant  Stimulus  Differences   1 1 1 

Let  us  write  L  for  DL^^  and  U  for  DL^^ ;  and  let  us  denote 
the  various  measures  of  precision  simply  by  h.  Then  we  have 
the  equations : 

/i  =  (i:  ±  A)^,  /i  =  (  ±  A-  U)h, 

t2=-{L±  D^)h,  4  =  ( ±  A-  ^)>^, 

t,  =  {L±D,)h',  t,  =  {±D,-U)h', 
which  must  be  worked  out  twice  over,  for  the  time  orders  I.  and 

II.     Referring  to   Fechner's  Fundamental   Table    (p.  99),  we 
find: 

1.22,79  =  {L  —  iso)k  o.oooo={—i^o—U)k 

0.762,9  =  (L— I  oo)h  0.0000  =  (— 100—  Ujh 

o.i428  =  (Z—  So)h  — 1.0994  =  (—  ^o—U)h 

— o.2i6o  =  (Z±     o)h  — 0.4549  =  (±     o—U)h 

— o.7639  =  (Z+   so)k  o.o7io  =  (+   ^o—U)k 

— i.0994  =  (Z4-ioo)>^  o.4i2i  =  (+ioo— ^)>4 

— 1.45  31  =  (Z  + 150)/^  i.0994-(+i5o— ^)^ 

o.7639  =  (Z-i5o)^  —1.453 1  =  (-150-6^)/^ 

0-595 1  =  (i^  — 100)/^  —0.8308  =  (— 100—  U)h 

—0.0355  =  (i:—  50)/^  —0.7639— (~  so--U)h 

— 0.4I2I  =  (Z±      o)>^  — o.29i7  =  (±      o—U)h 

— o.9936  =  (Z+   50)/^  o.io68  =  (+   50— ^)>^ 

— 1.4531  =  (Z+ioo)^  0.595 1  =  (+100—^)/^ 

0.0000  =  (Z+ 1 50)/^  0.7639  =  (+ 150— ^)/2 

If  we  weight  the  /-valves,  in  accordance  with  Miiller's  Table 

(p.  loi),  these  equations  become: 

i.2379(.047)  =  (Z-i50)>^ 

o.7639(-3iO  =  (^— ioo)>^ 

o.i428(.96o)  =  (Z—   so)h  —i. 09940089)  =  (—  50—6^)^ 

— o.2i6o(.9ii)  =  Z^  — o.4549(.66i)=  —   Uh 

— o.7639(.3ii)  =  (Z+   50)^  o.o7io(.99o)— (+   50— ^)>^ 

—  i.0994(.o89)  =  (Z  +  ioo)/^  o.4i2i(.7i2)  =  (+ 100— C/)^ 

—  i.453i(.oi4)  =  (Z+i5o)>^  i.0994(.o89)  =  (+i5o— ^).^ 


112  The  Metric  Methods 

Lu  ^„ 

o.7639(.3ii)  =  (^— 150)^  -i.453i(.oi4)  =  (-i5o-^)^ 

0-595 1  (492)  =  {L  —  ioo)h  — o.83o8(.25 1)  =  (—100—  U)h 

— 0.035 5 (.997)  =  (Z-  so)h  — o.7639(.3ii)  =  (-  so-U)h 

— 0.41 21  (.y  12)=^  Lh  — 0.291 7(. 843)=  —[/h 

— o.9936(.i39)  =  (Z+   50)^  o.io68(.977)  =  (+   50- 6^)/^ 

—  i.453i(.oi4)=(Z  +  ioo)/2  o.595i(.492)  =  (+ioo— C/)/2 

o.7639(.3ii)  =  (+i5o— C/)/^ 

From  these  we  obtain  the  normal  equations : 
[D^w]h—[Dw]DL.h=  [Dtzv], 
—  [Dw]h-\-[w]DL.h    =—[tw]; 

which  furnish  the  results : 

L,  =17.22  /2=.oo85  ; 

Z„  =  49.82  ;^  =  .oo94; 

U,  =48.38  >^  =  .oo94; 

6^^=28.40  ^=.0070. 

(i)  It  is  instructive  to  compare  these  values  with  the  approximative  results 
on  p.  109.     We  have: 

First  procedure :  30,       53,       43,       36; 

Second  procedure:         17.22,  49.82,  48.38,  28.40. 
The  discrepancy  between  the  two  figures  that  stand  for  the  Zj  is  startling ; 
but   it   is    accounted  for  by  the  fact  that  the  percentages  0.38  and  0.58  se- 
lected by  the  first  procedure  involve  a  very  distinct  inversion  of  the  second 
order. 

(2)  If  the  example  given  above  is  worked  out,  it  will  be  seen  that  the 
bracketed  sums  in  the  normal  equations  are  always  employed  as  positive 
values;  that  is  to  say,  the  signs  of  operation  (+,  — ,  +;  — ,  +,  — )  are 
retained,  no  matter  whether  the  sums  are  intrinsically  positive  or  negative. 
The  measure  of  precision  must,  of  course,  always  be  a  positive  quantity  (see 
p.  44  above).  Whether  the  £>L  be  positive  or  negative  depends,  not  on 
the  outcome  of  the  normal  equations,  but  on  the  character  of  the  original 
data.  Suppose,  e.  g., — the  illustration  is  taken  from  an  experiment  actually 
made, — that  we  obtained  results  like  these : 

n  -69.4  -53.8  -37.0  -19.5        o    +20.0  +39.7  +60.3  +80.7 

g         0.99       0.95       0.88       0.73  0.51        0.31        0.16       0.05       O.OI. 

Here  the  lower  DL  lies,  not  in  the  range  of  the  mmus-B^s  at  all,  but  some- 
where between  D  =  o  and  Z>=+2o!  The  normal  equations  for  the  data 
give  Z=  2.40,  ^  =  0.0200  ;  and  we  read  the  2.40  a.s  ^  minus  2.40^  because 
the  data  show  that  the  liminal  D  lies  on  the  wrong  side  of  the  D  =  o. 


§   22.    The  Method  of  Constant  Stimulus  Differences  113 

(4)  The  Fechnerian  Time  Error. — Our  four  Z>Z-values  are  all 
affected  by  the  time  error.  The  Fechnerian  time  error,  /,  is 
supposed  to  be  the  same  in  absolute  amount,  but  different  in 
sign,  in  the  two  time  orders.  It  is,  further,  termed  *  positive' 
when  its  effect  is  to  enhance  the  value  of  the  R  first  presented, 
*  negative '  when  its  effect  is  to  increase  the  subjective  value  of 
the  R  last  presented.  In  the  case  before  us,  a  positive  error 
would  accordingly  make  U^yU^^y  and  L^<iL^^.  This  is  what 
we  find. 

To  eliminate  /,  we  write  : 

i;?Z,=^^^i±^=  33.52  gr. 

To  determine  /  .* 

2p=U -U,,  or  L,-L,, 

/=+9-99    or  +16.30; 
av./=  + 13.145. 

(5)  Consideration  of  the  Results. — Are  these  results  satisfac- 
tory }  Let  us  see,  first  of  all,  whether  the  limens  conform  at  all 
closely  to  the  requirements  of  Weber's  Law.     We  have  : 

DU=        38-39  ■     . 


IO7I  27.89' 

DLi  =        33.52 


[071-33-52  30-95 
The  agreement  between  ^g-  and  ^  is,  perhaps,  all  that  we  could 
expect  in  view  of  the  small  number  of  the  experiments.  On  the 
other  hand,  the  /-values  are  suspicious ;  it  is  difficult  to  regard 
9.99  and  16.3  as  even  approximately  *  equal.'  Fortunately,  we 
have  a  very  simple  way  of  finding  out  whether  the  average  1 3. 145 
adequately  represents  the  time  error.  For,  in  terms  of  this  and 
of  the  other  averages  just  drawn, 

^i  =(33-52  —  13.145),  with  an  h  of  .0085  ; 

^ii  =  (33-52+ 13.145),  with  an  h  of  .0094; 

^i  =(38.39+13.145),  with  an  h  of  .0094; 

^11  =  (38-39—1 3- 145),  with  an  h  of  .0070. 
Now,  by  substituting  these  figures  for  the  symbols  in  the  right- 
hand  members  of  the  equations  on  pp.  11  if,  we  shall  obtain  a 

H 


114  ^^/^^  Metric  Methods 

series  of  /-values  whose  n  may  be  discovered  by  reference  to 
Fechner's  Fundamental  Table.  A  comparison  of  the  calculated 
with  the  observed  n  will  show  at  once  whether  our  mathematical 
procedure  has  been  adequate  or  inadequate.  The  results  of  the 
substitution  are  as  follows  : 


Z, 

^„ 

n  obs. 

n  cal. 

Diff. 

n  obs. 

n  cal. 

Diff. 

0.96 

0.94 

—.02 

0.86 

0.92 

+  .06 

0.86 

0.83 

-.03 

0.80 

0.76 

-.04 

0.58 

0.64 

+  .06 

0.48 

0.48 

±.00 

0.38 

0.40 

+  .02 

0.28 

0.27 

—.01 

0.14 

0.20 

+  .06 

0.08 

O.IO 

+  .02 

0.06 

0.07 

+  .01 

0.02 

0.03 

+  .01 

0.02 

0.02 

db.oo 

0.00 

0.0 1 

+  ,OI 

0.00 

0.0 1 

+  .01 

0.02 

0.04 

+  .02 

0.00 

0.02 

+  .02 

0.12 

O.I  I 

—.01 

0.06 

0.09 

+  .03 

0.14 

0.23 

+  .09 

0.26 

0.25 

—.01 

0.34 

0.40 

+  .06 

0.54 

0.51 

-.03 

0.56 

0.60 

+  .04 

0.72 

0.74 

+  .02 

0.80 

0.77 

-.03 

0.94 

0.90 

-.04 

0.86 

0.89 

+  .03 

The  discrepancies,  though  obvious  enough,  are  not  so  large  that 
they  may  not  be  accounted  for  by  the  small  number  of  experi- 
ments. 

So  far  as  our  analysis  has  gone,  therefore,  we  have  no  reason 
for  dissatisfaction.  We  set  out  from  a  small  group  of  data, — 
not  such  a  group  as  would  be  acquired  in  the  progress  of  an  in- 
vestigation, but  rather  such  a  group  as  might  be  obtained  in  a 
few  hours  of  laboratory  work.  Despite  the  small  number  of  the 
experiments,  we  have  found  upper  and  lower  (relative)  DL  which 
afford  a  rough  confirmation  of  Weber's  Law,  and  we  have  found 
a  value  for  the  positive  time  error  which  is  at  any  rate  approxi- 
mately representative  of  the  true  value.  It  should  go  without 
saying  that  these  results  presuppose  extreme  care  on  the  part  of 
Ey  and  extreme  conscientiousness  (together  with  some  prelimi- 
nary practice  in  the  judgment  of  weights)  on  the  part  of  O. 


§   22.    The  Method  of  Constant  Stimulus  Differences   115 

EXPERIMENT  XXH 

The  Determination  of  the  DL  for  Inte^tsity  of  Sound. — Ma- 
terials.— Sound  pendulum.     Soundless  metronome. 

Preliminaries. — E  is  to  choose  his  standard,  and  to  make 
out  a  list  of  7  variable  R.  One  of  these  ri  is  identical  with  r ; 
the  remaining  6  are  to  be  distributed  symmetrically  about  r, 
above  and  below.  Their  values  must  be  such  that  the  two  ex- 
treme ^1,  the  loudest  and  the  faintest  sounds,  are  hardly  ever 
confused  by  O  with  r,  the  standard  sound.  If,  e.  g.y  the  stand- 
ard r  is  given  by  a  fall  of  the  pendulum  through  45°,  the  series 
of  variables  might  perhaps  be  taken  as  60°,  55°,  50°,  45°,  39°, 
32°,  24°. 1  Each  of  these  r^  is  to  be  compared  with  r,  in  both 
time  orders,  50  times  over  ;  so  that  the  whole  number  of  com- 
parisons will  be  700.  E  should  make  out  a  plan  of  the  complete 
experiment  beforehand. 

The  limiting  values  of  r^  are  determined  in  preliminary  exper- 
iments, which  will  therefore  take  some  time,  and  require  to  be 
carefully  made.  Warming-up  series  must  be  given  at  the  begin- 
ning of  every  laboratory  session. 

EXPERIMENT    yyTTT 

The  Determination  of  the  DL  for  Lifted  Weights. — This  may 
be  regarded  as  the  classical  experiment  of  quantitative  psychology. 
On  the  psychophysical  side,  it  has  engaged  a  long  Hne  of  investi- 
gators :  Weber  himself,  Fechner  and  Hering,  all  employed  it  to 
test  the  validity  of  Weber's  Law ;  and  a  glance  at  the  current 
magazines  will  show  that  the  work  begun  by  them  has  continued 
down  to  the  present  day.  On  the  psychological  side  it  has  been 
made  by  L.  J.  Martin  and  G.  E.  Muller  the  vehicle  of  a  quali- 
tative analysis  of  the  sensory  judgment,  the  most  elaborate  and 
penetrating  that  we  have.  Hence  it  is  fitting  that  the  experi- 
ment should  have  been  reserved  to  the  last,  and  that  we  should 
now  approach  it  in  the  light  of  everything  that  we  have  learned, 
whether  as  E  or  as  (9,  from  the  preceding  experiments. 

Materials. — Fechner's  weight  holders,  with  set  of  weights. 
Uprights  with  tape.     Carrier  bracket.     Arm  rest.     Metronome, 

"^  E  can  set  the  releases  to  0.5°.  Hence,  if  this  series  is  actually  adopted,  E 
should  work  it  out  correctly. 


ii6 


The  Metric  Methods 


F1G.32. 


soundless  or  ordinary.  Screens.  [Fechner's  weight  holders 
consist  of  skeleton  cubes  of  brass  wire,  covered  with  a  solid  brass 
lid,  and  lifted  by  a  wooden  roller  handle  (Fig.  32).     To  the  lid  is 

soldered  a  small 
round  box  of  brass, 
itself  furnished 
with  a  cover.  The 
holders  may  be 
made  to  weigh, 
empty,  300,  400 
or  500  gr.,  accord- 
ing to  the  dimen- 
sions of  the  ma- 
terials. The  prin- 
cipal weights  are 
squared  slabs  of 
lead,  zinc,  etc.,  dif- 
fering in  thickness,  and  cut  to  lie  snugly  and  evenly  in  the  hold- 
ers. It  is  essential  that  there  be  no  rattling  or  shifting  of  the 
weights,  as  the  holder  is  lifted.  For  minor  variations  of  weight, 
discs  of  lead,  etc.,  may  be  dropped  into  the  round  boxes  carried 
by  the  lids. 

The  carrier  bracket  (Fig.  32)  is  a  device  for  the  elimination  of 
the  space  error.  Two  wooden  carriers,  turning  about  vertical 
axes,  are  so  connected  (by  the  rod  beneath  the  bracket)  that 
either  can  be  swung  into  the  place  vacated  by  the  other.  The 
carriers  are  faced  with  felt  or  baize,  for  the  reception  of  the 
holders  ;  their  movement  is  arrested  by  a  strip  of  felt-lined  wood 
placed  just  beneath  the  edge  of  0's>  table.  The  weight  holders 
are  set,  at  the  proper  angle,  upon  the  carriers ;  E  swings  them 
in,  alternately,  to  the  required  position  under  6^'s  hand :  O 
makes  the  double  lift  with  no  change  in  the  spatial  relations  of 
his  hand  and  arm.] 

Preliminaries. — There  are,  naturally,  various  ways  in  which 
the  experiment  may  be  performed.  With  the  arrangement  shown 
in  Fig.  32,  6>  is  supposed  to  be  seated  comfortably  at  the  side  of  a 
low  table,  his  forearm  lying  in  the  arm  rest  (plaster  mould  or  sand 
box),  and  his  hand  projecting  (back  upwards)  so  far  beyond  the 


§  22.   The  Method  of  Constant  Stimiibis  Differences    117 

edge  of  the  table  that  he  can  conveniently  grasp  the  handles  of 
the  weight  holders,  as  they  are  presented.  Two  upright  rods  are 
nailed  to  the  edge  of  the  table,  one  on  either  side  of  (9's  hand, 
and  a  piece  of  stout  tape  is  stretched  across  them,  at  a  height  of, 
say,  10  cm.     This  marks  the  height  of  lift. 

An  alternative  arrangement  is  that  O  stand  to  the  weights :  the  bracket 
is  fixed  at  such  a  height  that  he  can  easily  grasp  the  handle  with  his  hand, 
while  the  upper  arm  hangs  down  by  the  side  of  the  body  and  the  lower 
arm  (lying  in  a  plane  parallel  to  the  median)  makes  an  obtuse  angle  with  the 
upper.     The  height  of  lift  may  be  regulated  as  before. 

Fechner  allowed  i  sec.  for  raising,  i  sec.  for  lowering,  and  i  sec.  for 
changing  the  weight ;  so  that  each  observation  (double  lift)  required  5  sec. 
He  allowed  the  same  time,  5  sec,  to  elapse  between  observation  and  obser- 
vation. Whether  these  time  relations  are  kept  or  not  will  depend  upon  O's 
mode  of  lifting.  The  weights  may  be  lifted  evenly,  deliberately,  and  atten- 
tion paid  to  the  whole  course  of  the  up-down  movement ;  or  they  may  be 
picked  up,  as  one  picks  up  from  the  dining  table  an  empty  glass  that  one's 
neighbour  is  offering  to  fill, — lifted  all  of  a  piece,  with  more  or  less  of  a  jerk, 
at  the  hest  of  an  initial  impulse.  O  must  be  allowed,  after  practice,  to 
choose  his  own  manner  of  lifting  ;  if  he  decide  upon  the  second  mode,  the 
experiments  will  run  their  course  more  quickly  than  Fechner's. 

E  and  O  must  decide,  further,  whether  they  prefer  to  regulate  the  experi- 
ments by  a  ticking  or  a  soundless  metronome.  Sometimes  the  ticking 
serves  to  distract  the  attention  ;  sometimes  it  seems,  on  the  contrary,  to  help 
towards  a  better  concentration  of  attention,  the  metronome  taking  upon 
itself  (so  to  say)  a  part  of  the  responsibility  for  the  conduct  of  the  ex- 
periment. 

The  screens  are  so  arranged  that  O  can  see  nothing  of  E'^%  operations 
with  the  weights. 

Having  chosen  his  standard  weight,  E  determines,  in  prelimi- 
nary experiments,  the  limiting  values  of  ^i,  above  and  below  r. 
The  determination  may  take  some  little  time,  and  must  be  care- 
fully made.  Eight  weight  holders  are  then  prepared  (standard, 
and  7  variable  weights),  and  marked  by  E  with  conventional 
signs.  E  makes  out  a  plan  of  the  complete  experiment  be- 
forehand. 

There  is  no  reason  why  O  should  look  at  the  weight  holders  at  all,  after 
the  first  practice  series  have  been  taken.  He  may,  however,  chance  to  see 
them  ;  and  it  is  important  that,  if  he  does,  they  shall  present  no  recog- 
nisable differences. 


Ii8  The  Metric  Methods 

Every  r-^  is  to  be  compared  with  r  50  times,  in  both  temporal  orders. 
These  700  comparisons  give  plenty  of  chance  for  confusion  :  much  more 
than  the  corresponding  700  with  the  sound  pendulum,  where  the  r  and  r^  are 
not  separate  things,  but  simply  different  settings  of  a  stationary  instrument. 
Hence  it  is  important  for  E  (i)  to  make  out  his  programme  fully  and  clearly, 
series  by  series  ;  (2)  to  arrange  his  weight  holders  in  a  certain  fixed  order, 
indicated  by  labels  pasted  on  the  table  (the  labels  showing  both  the  actual 
weight  of  the  holder  and  its  contents,  and  also  the  conventional  sign  with 
which  the  particular  holder  has  been  marked) ;  and  (3)  to  move  the  carriers 
always  in  a  certain  order  (right  first,  left  second),  and  to  dispose  accordingly 
the  weights  concerned  in  a  particular  comparison. 

Warming-up  series  must  be  given  at  the  beginning  of  every 
laboratory  session. 

Questions. — E  and  O  {\)  Write  out  in  brief  the  procedure  of 
the  Method  of  Constant  Stimulus  Differences,  distinguishing 
carefully  the  successive  stages  in  the  procedure.  What  is  the 
psychology  of  the  method  .? 

(2)  We  have  done  nothing,  in  our  illustration  of  the  method, 
with  the  judgments  of  *  much  greater '  and  *  much  less.'  To  what 
psychophysical  account  may  these  judgments  be  turned } 

(3)  In  the  Table  on  p.  107  we  reversed  the  judgments  given  in 
the  first  time  order  (5  lifted  first),  in  order  that  all  judgments 
alike  might  refer  to  5.  Is  this  reversal  a  simple  matter  of 
course  }  Or  has  it  any  psychological  implications  }  How  would 
you  seek  to  show  that  it  is  psychologically  permissible } 

(4)  What  is  the  psychological  ground  or  introspective  basis  of 
the  ^^-judgment .? 

(5)  What  advantage  is  there  in  using  a  number  of  positive 
and  negative  Z>'s  for  the  determination  of  the  DL  ?  Could  not 
the  DLu  be  determined  by  the  aid  of .  positive  U  s  only  .?  And 
would  not  the  results  obtained,  say,  with  two  such  Hs  show  with 
sufficient  accuracy  whether  Weber's  Law  holds  or  does  not  hold 
for  the  sense  department  under  investigation  ? 

(6)  Criticise  Fechner's  view  of  the  constant  errors  of  time  and 
space. 

(7)  Discuss  the  choice  of  the  five  forms  of  judgment :  'much 
greater,*  *  greater,'  *  doubtful',  Mess,'  'much  less.' 

(8)  We  said  on  p.  106  :  "  the  5  is  to  be  compared  with  each  of 
the  Cy  taken  in  haphazard  order y  50  times  over."     What  other 


§  22.    The  Method  of  Constant  Stimulus  Differences    119 

arrangements  of  the  experiment  are  possible  ?  What  are  their 
respective  advantages  and  disadvantages  ? 

(9)  We  also  said  on  p.  106  :  "  6>  judges  always  in  terms  of  the 
weight  lifted  second."  What  is  the  reason  for  this  rule?  Why 
should  not  O  judge  in  terms  of  C,  or  in  terms  of  5,  irrespective 
of  the  time  order  ?  Why  should  he  not  be  left  entirely  free  to 
judge  as  he  will  from  one  comparison  to  another  ? 

( 10)  The  C^  of  p.  106  lie  symmetrically  to  5,  above  and  below. 
Is  this  the  best  disposition  of  the  C^  ?     Why? 

(11)  What  are  your  criteria  of  judgment  in  the  comparison  of 
lifted  weights  ? 

(12)  Can  you  suggest  an  improvement  of  the  apparatus? 


CHAPTER    III 

THE    REACTION    EXPERIMENT 

§  23.  The  Electric  Current  and  the  Practical  Units  of  Electri- 
cal Measurement. — (i)  When  an  insulated  substance  is  charged 
with  electricity,  it  becomes  capable  of  doing  work ;  in  other 
words,  the  process  of  charging  is  accompanied  by  the  accumula- 
tion of  potential  energy  in  the  substance.  In  technical  terms,  we 
say  that  the  charged  substance  has  a  certain  potential.  If,  now, 
we  connect  by  a  conductor  two  bodies  of  different  potential, 
there  is  produced  in  the  conductor  what  is  called  an  electric  cur- 
rent. There  is  a  strain  or  tension  between  the  two  points  con- 
nected, and  the  electric  current  tends  to  transfer  this  strain 
through  the  conductor.  A  difference  of  potential  or  D.  P.  be- 
tween two  points  may  therefore  be  defined  as  that  difference  in 
electrical  condition  which  tends  to  produce  a  transference  of  elec- 
tricity from  the  one  point  to  the  other.  We  say,  conventionally, 
that  the  electric  current  flows  from  the  body  at  the  higher  poten- 
tial to  the  body  at  the  lower  potential. 

While,  however,  a  D.  P.  is  able  to  produce,  it  is  not  able  con- 
tinuously to  maintain  an  electric  current.  If  the  current  is  to  be 
maintained,  the  D.  P.  itself  must  be  maintained.  The  force  or 
agency  which  keeps  up  a  permanent  D.  P.  in  any  electric  gener- 
ator, primary  battery  or  what  not,  is  termed  electromotive  force 
or  E.  M.  F.  In  practice,  it  is  seldom  necessary  to  make  the 
distinction  between  E.  M.  F.  and  D.  P.  Thus,  for  purposes  of 
measurement,  the  E.  M.  F.  of  a  cell  is  taken  as  equal  to  the 
maximal  D.  P.  between  the  terminals  of  the  cell  on  open  circuit. 

The  practical  unit  of  E.  M.  F.  or,  as  it  is  generally  termed,  of 
electrical  pressure,  is  the  volt.  A  gravity  cell,  such  as  that  ordi- 
narily used  with  a  telegraph  instrument,  gives  an  E.  M.  F.  of 
about  one  volt. 

(2)  Every  conductor  offers  a  certain  resistance  to  the  passage 
of  an  electric  current,  the  amount  of  resistance  varying  inversely 
with  its  size  and  directly  with  its  length,  and  differing  also 
with  the  material  of  which  it  is  made.     The  unit  of  resistance  is 


§  23-   The  Electric  Current  and  Units  of  Measurement   121 

termed  the  ohm.  The  copper  wire  ordinarily  employed  for 
electric  lighting  in  houses,  no.  14  Brown  and  Sharp  gauge,  has 
at  summer  temperature  a  resistance  of  0.00259  ohm  per  foot. 

(3)  We  have  lastly  to  consider  the  current  itself, — the  current 
which  flows  along  the  conductor  under  a  certain  pressure  (meas- 
ured in  volts)  against  a  certain  resistance  (measured  in  ohms). 
Current  is  the  rate  of  flow  of  electricity.  The  practical  unit  of 
current  strength,  or  the  unit  rate  of  flow  of  electricity,  is  the 
ampere.  An  ordinary  16  c.  p.  no  volt  incandescent  lamp 
requires  a  current  of  about  half  an  ampere. 

(4)  These  three  fundamental  factors,  pressure,  current  and 
resistance,  are  present  in  every  active  electrical  circuit.  The  flow 
of  electricity  along  a  conductor  thus  presents  a  fairly  definite 
analogy  to  the  flow  of  water  through  a  pipe.  When  a  reservoir 
supplies  a  city  with  water  through  a  long  pipe,  the  rate  of  flow  at 
the  orifice  of  the  pipe  depends,  first,  upon  the  pressure  which 
drives  the  liquid  through  the  pipe,  i.  e.,  upon  the  height  of  the 
water  in  the  reservoir  above  the  outlet,  and,  secondly,  upon  the 
length  and  size  of  the  pipe.  The  quantity  of  water  discharged 
in  the  i  sec,  or  the  rate  of  flow,  here  corresponds  to  the  rate  of 
flow  of  the  electric  current,  measured  in  amperes  ;  the  friction  in 
the  pipe  corresponds  to  the  resistance  of  the  conductor,  measured 
in  ohms ;  the  relationship  between  E.  M.  F,  and  D.  P.y  both 
measured  in  volts,  is  the  same  as  the  relationship  between  force 
of  gravity  and  water  pressure  or  difference  of  water  level. 

(5)  It  is  usual  to  denote  the  E.  M.  F.y  current,  and  resistance 
of  an  active  circuit  by  the  letters  E,  I  and  R  respectively.  ^  The 
connection  between  them  is  expressed  in  what  is  known  as  Ohm's 
Law,  which  asserts  that 

i.  e.y  that 

voltage 
amperage  =  ^^; 

in  words,  that  the  strength  of  the  current  varies  directly  as  the 

1  At  the  Internat.  Elec.  Congress  of  1893  i*  was  decided  that  'current'  should  be 
denoted  by  /  (intensity)  instead  of  by  C.  The  latter  symbol  is,  however,  still  found 
in  many  current  editions  of  works  upon  electricity. 


122 


TJie  Reaction  Experiment 


electromotive    force  and  inversely  as  the  total  resistance  of  the 

circuit.     It    follows,  of  course,  that   E  =  I X  R,  and  R  =  -  -.      If, 

then,  any  two  of  the  three  factors  are  known,  we  can  at  once  cal- 
culate the  third.  Suppose,  e.  g.,  that  we  have  2i  D.  P.  oi  lo 
volts,  and  a  total  resistance  of  5  ohms  in  circuit :  then  we  shall 
get  a  current  of  2  amperes.  Suppose  that  we  desire  to  send  a 
current  of  100  amperes  through  a  resistance  of  2  ohms  :  we  shall 
need  a  pressure  of  200  volts  to  furnish  it.  Suppose,  finally,  that 
we  have  a  i).  P.  of  50  volts,  and  that  a  current  of  2  amperes  is 
traversing  the  circuit :  the  resistance  of  the  total  circuit  must  be 
2$  ohms. 

The  Voltaic  Cell. — (i)  Voltaic  cells  exist  in  a  great  variety  of 
forms  ;  but  in  all  three  parts  are  present, — two  conducting  plates, 

called  the  elements,  and  forming  to- 
gether a  voltaic  couple ;  and  a  liquid 
surrounding  the  plates,  called  the 
electrolyte.  A  simple  cell  may  be 
made  by  putting  a  strip  of  copper 
and  a  strip  of  zinc — which  must  not 
touch  each  other — into  a  glass  jar 
containing  dilute  sulphuric  acid.  If 
such  a  cell  be  put  on  closed  circuit, 
i.  e.y  if  the  copper  and  zinc  ter- 
minals be  connected  by  a  copper 
wire,  a  continuous  current  of  elec- 
tricity will  flow  through  the  wire 
from  the  copper  to  the  zinc.  The  copper  terminal  is  thus  the 
positive,  the  zinc  the  negative  pole  of  the  cell.  The  circuit  is 
completed,  within  the  jar,  by  the  flow  of  current  through  the 
liquid  from  the  zinc  to  the  copper  strip ;  so  that  the  zinc  is  the 
positive,  the  copper  the  negative  plate  or  element.  The  E.  M.  F. 
of  such  a  cell  is  measured,  as  we  have  said,  by  the  D.  P.  between 
the  positive  and  the  negative  poles. 

The  simple  cell,  as  described,  has  two  drawbacks  :  polarisation 
and  local  action.  Polarisation  takes  place  when  the  bubbles  of 
hydrogen  liberated  at  the  surface  of  the  copper  plate  adhere  to 
it,  and  so  form  a  resistant  film  which  weakens  the  current.    Local 


Fig.  33.     Simple  voltaic  cell. 


§  23.    The  Electric  Current  and  Units  of  Measnrement   123 

action  takes  place  when  particles  of  iron,  carbon,  etc.,  present  as 
impurities  in  the  zinc  plate,  form  separate  voltaic  circuits  with 
the  zinc  itself,  thus  acting  as  the  negative  elements  of  a  number 
of  little  voltaic  cells  ;  energy  is  hereby  diverted  from  the  main 
circuit,  and  the  chemicals  are  quickly  wasted.  Polarisation  is 
avoided  by  various  means,  which  are  detailed  in  the  larger  works 
on  physics  and  in  the  special  books  on  primary  batteries  {e.g., 
H.  S.  Carhart,  Primary  Batteries,  1891);  local  action  is  pre- 
vented by  amalgamation  of  the  zinc  with  mercury. 

(2)  For  laboratory  purposes,  cells  may  be  divided  into  two 
classes  :  those  that  should  remain  normally  on  open  circuit,  and 
those  whose  circuit  should  normally  remain  closed.  Cells  of  the 
former  class  are  intended  for  use  only  for  a  few  minutes  at  a  time  ; 
cells  of  the  latter  class  may  be  used  continuously,  until  they  are 
practically  exhausted.  Open  circuit  cells,  of  which  the  Leclanch6 
cell  is  typical,  will  probably  always  be  employed  in  laboratories 
for  the  ringing  of  signal  bells,  for  short-distance  telephones,  etc. 
Closed  circuit  cells,  of  which  the  Daniell  is  typical,  may  be  em- 
ployed for  running  small  motors,  for  actuating  the  Hipp  chrono- 
scope,  etc.,  though  in  most  laboratories  they  are  being  replaced 
by  storage  batteries  or  by  direct  current  service  from  the  univer- 
sity or  city  lighting  plant. 

The  open  circuit  cell  must  {a)  be  capable  of  immediate  response  when  the 
circuit  is  closed,  without  previous  preparation  and  for  months  at  a  time  \  and 
{b)  must  depolarise  when  left  to  itself.  In  the  Lechanch^  cell,  the  positive  plate 
is  a  rod  of  zinc;  this  is  immersed  in  liquid  (an  aqueous  solution  of  ammonium 
chloride)  which  acts  upon  it  only  when  the  circuit  is  closed.  The  negative 
plate  is  a  hollow  cylinder  composed  of  broken  carbon  and  of  dioxide  of 
manganese,  which  latter  acts  as  a  slow  depolariser.  The  cell  has  an^.  M.  F. 
of  about  1.5  volts;  its  internal  resistance  is  high  (as  might  be  gathered 
from  the  small  size  of  the  zinc  rod),  varying  from  0.4  to  2  ohms. 

The  closed  circuit  cell  must  furnish  a  strong  continuous  current  over  a  con- 
siderable interval  of  time.  The  conditions  to  be  met,  therefore,  are  those 
of  {a)  low  internal  resistance  and  {b)  prompt  depolarisation.  We  have  already 
referred  to  the  Daniell  as  a  typical  cell  of  this  class.  The  cell  is  composed 
of  a  zinc-copper  couple,  separated  by  a  porous  partition  ;  the  zinc  plate  is 
immersed  in  dilute  sulphuric  acid,  the  copper  plate  in  a  saturated  solution  of 
sulphate  of  copper.  During  action,  the  copper  plate  receives  an  electrolytic 
deposit  of  metallic  copper.  As  ordinarily  set  up,  the  Daniell  cell  has  an  E.. 
M.  F.   of  1.08  volts,  and  an  internal  resistance  of  0.85  ohm.     The  internal 


124 


The  Reaction  Experiment 


resistance  is  thus  rather  large  in  comparison  with  the  E.  M.  F.j  only  a 
moderate  current,  about  an  ampere,  can  be  taken  from  a  Daniell  as  a  maxi- 
mum. On  the  other  hand,  the  current  is  extremely  constant.  This  was 
the  cell  used,  e.  g.,  by  Wundt  and  Dietze  in  their  investigation  of  the  range 
of  consciousness  ;  it  is  figured  by  Wundt  in  the  Physiol.  Psychologic,  iii., 
1903,  361.  The  cells  figured  ibid.,  i.,  1902,  512  are  Meidinger  cells, — 
gravity  cells  of  special  construction,  much  used  in  Germany,  with  zinc  in  a 
solution  of  magnesium  sulphate  and  copper  in  a  solution  of  copper  sulphate. 
Another  well-known  form  of  the  closed  circuit  cell  is  the  Grenet  or  bichro- 
mate cell.  In  this,  the  negative  plate  (with  positive  pole  !)  consists  of  two 
parallel  slabs  of  carbon,  metallically  connected  at  the  top  ;  the  positive  plate 
(negative  pole)  is  a  plate  of  zinc,  lying  between  the  carbons.  The  large 
surfaces  of  the  plates  and  their  close  proximity  secure  a  very  low  internal 
resistance.  The  liquid  is  dilute  sulphuric  acid  in  which  potassium  bichro- 
mate has  been  dissolved.  The  chromic  acid  formed  in  the  solution  gives  up 
a  part  of  its  oxygen  to  the  hydrogen  as  fast  as  the  latter  appears  ;  it  thus 
acts  as  a  very  rapid  depolariser.  The  initial  E.  M.  F.  of  the  bichromate 
cell  is  almost  twice  as  high  as  that  of  the  Daniell :  about  1.9  to  2.1  volts.  If 
furnished  with  a  zinc  plate  4  by  3  in.,  it  will  yield  a  current  of  2  amperes. 
The  current  is,  however,  less  constant  than  that  of  the  Daniell. 

Practical  rules  for  the  care  and  use  of  these  cells  must  be  learned  from 
the  special  handbooks. 

(3)  If  a  number  of  conductors  of  equal  conductivity  be  joined 

end  to  end,  so  as  to  form  one 


z 

V 

. 

-t- 

-' 

-t- 

z 

z 

SEFUES. 


o    p 


long  conductor,  the  resistance 
offered  to  the  flow  of  a  current 
will  evidently  increase  as  the 
number  of  individual  conduc- 
tors. If  the  same  conductors 
are  placed  side  by  side,  so  as  to 
form  a  multiple  bridge  between 
the  points  of  highest  and  lowest 
pressure,  the  resistance  will  evi- 
dently fall  in  proportion  to  the 
number  of  the  conductors  .so 
placed.  The  former  arrange- 
ment is  known  as  arrangement 
in  series,  the  latter  as  arrange- 
ment in  parallel. 

These   considerations    apply 
to  the  formation  of  batteries  by  the  connection   of   like    cells. 


PARALLEL. 

- 

+ 

?           : 

+ 

c 

; _ 

2  :  z        ' 


Fig.  34. 


;3.    The  Electric  Current  and  Units  of  Measurement   125 


First,  let  the  cells  be  connected  in  series :  the  positive  terminal 
of  one  to  the  negative  terminal  of  another,  the  positive  terminal 
of  this  to  the  negative  of  a  third,  and  so  on.  The  battery  gives 
an  increase  of  E.  M.  F.,  since  the  E.  M.  F.  of  the  first  cell  is 
added  to  that  of  the  second,  and  so  on  throughout  the  series. 
But  there  is  also  an  increase  of  internal  resistance,  by  a  similar 
addition  of  the  internal  resistances  of  the  separate  cells.  Hence, 
if  n  represents  the  number  of  cells  in  the  battery,  r  the  internal 
resistance  of  a  single  cell,  and  R  the  resistance  of  the  external 
circuit,  we  have 

T—   ^  ^ 

~//r+R' 

Whether,  now,  we  have  increased  our  available  current  depends 
entirely  on  the  value  of  R.  If  R  is  so  small  as  to  be  negligible 
in  comparison  with  nVy  we  have,  approximately, 

y  ;zE  E  ^ 
nr  r  * 
under  such  conditions,  an  infinite  number  of  cells  in  series  cannot 
maintain  a  larger  current  than  a  single  cell  on  short  circuit.  If, 
on  the  other  hand,  r  is  negligible  in  comparison  with  R,  increase 
in  the  number  of  cells  increases  the  current  in  nearly  the  same 
ratio.     For  in  that  case  we  have,  approximately, 

T     «E        E 

Secondly,  let  the  cells  be  connected  in  parallel  :  the  positive 
terminals  are  all  joined  together  to  form  the  positive,  the  nega- 
tive terminals  to  form  the  negative  terminal  of  the  battery.  The 
E.  M.  F.  remains  unchanged  ;  but  the  internal  resistance  falls  to 

-th  of  that  of  the  single  cell.     We  therefore  obtain 


-+R 

n 

What  we  have  done  again  depends  entirely  upon  the  value  of  R. 

If  R  is  negligible  in  comparison  with  r,  or  even  with  -,  then 

T        E  E 

1  = =  «-, 

r  r 

n 

or  the  current  is  «-times  greater  than  can  be  taken  from  the 


126  The  Reaction  Experiment 

single  cell.  But  as  R  increases,  the  gain  decreases  :  until  at 
last,  if  R  is  equal  to  or  greater  than  -,  there  is  no  apprecia- 
ble advantage  in  adding  to  the  number  of  cells  in  the  battery. 

In  a  word,  then  :  to  increase  our  current  in  face  of  a  low  ex- 
ternal resistance,  we  must  join  up  a  number  of  cells  in  parallel; 
to  increase  it  in  face  of  a  high  external  resistance,  we  must  join 
up  a  number  of  cells  in  series.  We  might  accomplish  the  same 
end,  in  the  first  instance,  by  increasing  the  size  of  the  cell  and  so 
decreasing  its  internal  resistance,  and  in  the  second  by  increas- 
ing the  E.  M.  F.  of  the  cell.  Since,  however,  the  enlarging  of 
the  single  cell  would  soon  make  it  unwieldy,  and  since  there  is 
no  known  cell  whose  E.  M.  F.  exceeds  3  volts,  we  have  recourse 
in  practice  to  the  formation  of  batteries  from  small  cells. 

We  may  illustrate  our  formulae  by  some  numerical  instances.  Suppose 
that  we  have  a  cell  whose  E.  M.  F.  is  2  volts,  and  whose  internal  resistance 
is  o.i  ohm.  Such  a  cell  will,  by  Ohm's  law,  give  20  amperes  of  current  on 
short  circuit ;  it  will  give  16.6  amperes  on  a  circuit  whose  external  resis- 
tance is  0.2  ohm ;  it  will  give  0.327  ampere  on  a  circuit  whose  external 
resistance  is  6  ohms. 

Suppose  that  6  of  these  cells  are  connected  in  series.  On  short  circuit, 
the  E.  M.  F.  of  the  battery  is  12  volts,  the  internal  resistance  0.6  ohm. 
The  current  is  20  amperes,  as  with  the  single  cell.  If  the  external  resis- 
tance is  6  ohms,  the  E.  M.  F.  of  the  battery  is  12  volts,  the  total  resistance 
6,6  ohms.  The  current  is  now  1.81  amperes,  as  against  the  0.327  of  the 
single  cell. 

Suppose  that  6  of  the  cells  are  connected  in  parallel.  The  E.  M.  F. 
remains  unchanged ;  but  the  internal  resistance  of  the  battery  is  only  0,016 
ohm.  With  an  external  resistance  of  0.02  ohm,  the  current  in  the  circuit  is 
55.5  amperes,  as  against  the  16,6  of  the  single  cell.  With  an  external 
resistance  of  6  ohms,  the  current  is  0,332  ampere,  or  practically  the  same  as 
that  of  the  single  cell. 

These  two  methods  of  connecting  cells  may  be  variously  combined,  and 
the  E.  M.  F.  and  internal  resistance  of  the  battery  may  thus  be  variously 
adjusted.  A  battery  composed  of  n  cells  arranged  in  m  series  with  /  cells 
in  each  series  is  said  to  be  grouped  in  tnultiple-series.  For  a  steady  cur- 
rent of  maximal  value,  the  cells  should  be  so  joined  up  that  the  internal 
resistance  of  the  battery  is  as  nearly  as  possible  equal  to  the  resistance  of 
the  external  circuit. 

Storage  Batteries  or  Accumulators. — A  storage  battery  is  an 
appliance,  not   for  the  storage  of   electricity, — which  cannot  be 


§  23-    TJie  Electric  Current  unci  Units  of  Measurement   127 

stored, — but  for  the  storage  of  energy  which  is  delivered  to  it  in 
the  form  of  electricity,  and  which  it  will  return  in  the  same 
form. 

The  simplest  type  of  storage  battery  consists  of  two  sets  of 
lead  plates  immersed  in  dilute  sulphuric  acid.  If  a  current  of 
electricity  is  sent  through  it,  for  a  certain  length  of  time, 
peroxide  of  lead  is  deposited  upon  the  plates  by  which  the  cur- 
rent enters  the  cell,  while  the  other  plates  become  spongy 
metallic  lead.  This  process  constitutes  the  charging  of  the  bat- 
tery. When  the  charging  is  completed,  the  peroxide  or  brown 
plates  are  the  positive,  the  spongy  lead  or  grey  plates  the  neg- 
ative plates  of  the  cell :  if  the  poles  are  connected  by  a  con- 
ductor, a  current  will  be  given,  precisely  as  in  the  case  of  ,the 
primary  battery.  When  the  storage  battery  is  exhausted,  it  is 
recharged  by  a  current  sent  through  it  from  the  negative  to 
the  positive  pole  ;  ^  the  discharging  current  always  flows  in  the 
opposite  direction  to  that  of  the  charge. 

Storage  batteries  may  be  used  in  the  laboratory  for  all  pur- 
poses for  which  batteries  of  closed  circuit  primary  cells  can  be 
employed.  In  general,  they  are  less  bulky,  more  economical, 
and  cleaner  than  primary  batteries.  The  efficiency  of  the 
best  types,./.  ^.,  the  ratio  of  the  energy  stored  in  charging  to  the 
energy  given  out  as  the  battery  sinks  back  to  its  original  con- 
dition, is  from  70  to  80%;  but  depreciation  is  very  rapid  if  the 
battery  is  not  given  proper  care.  Full  instructions  as  regards 
time  of  charging,  rate  of  discharge,  care  of  cells,  etc.,  are  given 
with  all  purchasable  batteries. 

Storage  batteries  are  best  charged  from  a  direct  line ;  the  proper  resis- 
tance may  be  inserted  into  the  circuit  in  the  shape  of  incandescent  lamps. 
Since  the  charging  current  must  be  sent  through  the  battery  in  a  certain 
direction,  and  since  the  dynamo  may  at  any  time  have  been  reversed  with- 
out one's  knowledge,  it  is  necessary  to  test  the  polarity  of  the  posts  to  which 
the  battery  is  to  be  attached.  Pole  testers  of  various  kinds  are  on  the 
market.  But  the  +  and  —  wires  may,  perhaps,  be  most  simply  identified  by 
aid  of  a  pocket  compass.  One  of  the  effects  of  the  electric  current  is  that 
it  evinces  magnetism  tangentially  to  its  flow ;  that  is  to  say,  a  live  wire  is 
magnetic  at  right  angles  to  the  flow  of  the  current.     Stretch  a  wire  between 

1  In  other  words,  the  positive  wire  of  the  source  of  supply  must  be  led  to 
the  positive  pole  of  the  storage  battery. 


128 


TJie  Reaction  Experiment 


the  posts  that  constitute  the  terminals  of  the  direct  line,  and  hold  the  com- 
pass beneath  the  wire  in  such  manner  that  the  north  pole  of  the  needle 
would  naturally  point  towards  you.  If  the  current  is  flowing  from  you,  the 
north  end  of  the  needle  is  deflected  to  your  left, — as  if  the  wire  had  along 
its  lower  surface  a  bar  magnet,  at  right  angles  to  it,  with  the  south  pole  to 
the  left.  If  the  current  flows  toward  you,  the  north  end  of  the  needle  goes 
to  your  right.  Conversely,  if  the  compass  be  held  above  the  wire,  the  north 
end  of  the  needle  turns  to  the  right  when  the  current  flows  from  you,  to  the 
left  when  it  flows  towards  you.  In  general :  think  of  yourself  as  swimming 
in  the  direction  of  the  current,  with  yonr  face  turned  to  the  needle ;  then  the 
north  pole  of  the  needle  is  in  every  case  deflected  towards  your  left  hand. 

The  Distribution  of  Current  through   the  Laboratory . — Wet 
batteries,  whether  primary  or  secondary,  have  their  objectionable 


Nichols  rheostat. 


features.^  Their  use  may  be  avoided  if  the  laboratory  is  suppHed 
by  a  direct  line  from  the  university  or  city  power  plant.  We 
will  suppose  that  the  current  delivered  is  the  i  lo  volt  direct 
current,  and  that  its  available  maximum  has  been  limited  by  fuse 
plugs  to  some  15  or  20  amperes.  We  will  suppose  also  that  the 
laboratory  consists  of  one  large  room,  and  that  the  current  must 
be  distributed  to  several  pairs  of  students,  working  at  different 
parts  of  it.     The  distribution  may  be  effected,  most  simply,  by 

1  For  open  circuit  work,  the  Lechanche  cells  described  above  may  well  be  re- 
placed by  '  dry '  cells,  which  are  light,  cheap  and  clean. 


§  23.    The  Electric  Current  and  Units  of  Measurement   129 

means  of  a  Nichols  tinned  iron  rheostat,  or  of  a  set  of  Wright- 
Scripture  lamp  batteries. 

(i)  To  construct  the  Nichols  rheostat,  take  four  or  six  full-sized  sheets  of 
tinned  iron,  and  slit  them  nearly  through  from  opposite  sides  at  distances  of 
about  I  cm.  Mount  the  zigzag  strips  thus  obtained  upon  a  wooden  frame,  as 
shown  in  Fig.  35.  Connect  the  free  ends  of  each  strip  to  a  binding  post 
screwed  into  the  frame,  and  join  up  all  but  the  terminal  posts  by  pieces  of 
heavy  copper  wire,  so  that  the  four  or  six  strips  represent  one  continuous 
strip.  The  rheostat  may  now  be  inserted  between  the  terminals  of  the  di- 
rect line ;  the  connecting  wires  may  be  so  arranged  that  the  whole  frame  of 
four  or  six  sheets  is  in  the  circuit,  or  that  only  one,  two,  etc.,  sheets  are  in- 
cluded. The  current  strength  will  be  diminished  in  proportion  to  the  amount 
of  resistance  (length  of  tinned  iron  strip)  introduced. 

In  a  simple  circuit,  the  current  strength  is  the  same  at  all  points  of  the 
circuit.  Suppose,  e.  g.^  that  the  resistance  we  have  put  into  the  main  circuit 
has  reduced  the  current  strength  to  5  amperes.  Then  if  we  attach  to  the 
rheostat  a  pair  of  wires  leading  to  an  apparatus,  we  shall  get  a  cur- 
rent strength  of  5  amperes  in  the  branch  circuit,  no  matter  whether  the 
length  of  strip  included  between  the  branch  wires  is  long  or  short.  On  the 
other   hand,  the  E.  M.   F.   of   a  simple  circuit  is   uniformly  distributed 

E 
throughout  that  circuit.     If  I  =  — ,  and   I   is  constant,  any  variation  of  R 

must  be  accompanied  by  a  corresponding  variation  of  E.  Or,  concretely : 
if  we  have  a  constant  current  flowing  along  a  wire  of  constant  cross-section, 
and  we  tap  this  wire  at  different  points,  the  drop  in  volts  along  the  wire 
must  be  uniform  and  proportional  to  the  resistance  between  the  points.  If, 
with  the  whole  frame  in  circuit,  we  get  a  current  of  5  amperes  under  a 
pressure  of  no  volts,  then  with  half  the  frame  in  circuit  we  must  get  a 
current  of  5  amperes  under  a  pressure  of  55  volts,  with  a  tenth  of  the  frame 
in  circuit  a  current  of  5  amperes  under  a  pressure  of  1 1  volts  and  so  on.i 
We  take  advantage  of  this  law,  for  the  purposes  of  distribution  of  current,  in 
the  following  way.  The  two  or  three  longitudinal  strips  of  wood  at  the 
top  of  the  rheostat  frame  are  grooved.  In  the  grooves  run  sliding  blocks  of 
wood,  which  carry  a  binding  post  screwed  down  upon  a  strip  of  sheet  copper  ; 
the  end  of  the  strip  is  bent  down  and  over,  so  that,  as  the  slider  is  pushed 
along,  coiitact  is  made  at  cm.  intervals  with  the  successive  strips  of  tinned 
iron.  The  wires  running  to  the  apparatus  are  now  connected,  the  one  with 
the  binding  post  of  a  slider,  the  other  with  one  of  the  binding  posts  fixed  in 
the  side  of  the  frame  and  constituting  the  terminal  of  one  of  the  separate 
tinned  iron  sheets.  We  thus  get,  for  our  apparatus  circuit,  a  current  of  5 
amperes  under  a  pressure  of  5,  10,  15,  etc.,  volts,  the  voltage  varying  di- 
rectly with  the  proportional  distance   between    the   binding  posts  of   the 

1  The  strip  cut  as  described  in  the  text  from  a  single  sheet  of  tinned  iron  offers, 
in  rough  average,  a  resistance  of  2  ohms  to  the  passage  of  the  electric  current. 
I 


130  The  Reaction  ExperUnent 

rheostat.  Half-a-dozen  such  branch  circuits,  of  various  voltages,  may  be 
taken  off  from  the  one  frame.  Their  current  strength  may,  of  course,  be 
still  further  reduced,  where  necessary,  by  the  insertion  of  resistance  in  series 
with  the  apparatus. 

Handles  should  be  attached,  for  convenience  of  lifting,  to  the  uprights 
at  the  ends  of  the  frame.  Care  must  be  taken  that  the  tinned  iron  strip 
does  not  become  unduly  heated  ;  though  in  the  author's  experience  no 
danger  has  ever  arisen  from  this  source. 

The  Nichols  frame  has  other  uses  in  the  laboratory.  Placed  out  of  the 
way,  under  the  demonstration  table  of  the  lecture  room,  it  may  be  connected 
up  as  resistance  with  the  motor  that  drives  a  demonstration  colour  mixer, 
or  with  the  arc-light  projection  lantern.  The  arc-light,  e.  g.^  requires  a  cur- 
rent of  5  to  10  amperes  under  a  pressure  of  40  to  50  volts.  Suppose  that 
we  have  at  the  terminals  of  the  direct  line  a  1 10  volt  20  ampere  current.  This 
means  that  the  resistance  in  the  wires  leading  to  the  dynamo  is  5.5  ohms.  By 
means  of  the  rheostat,  we  introduce  into  the  circuit  a  further  resistance  of 
5.5  ohms.  A  pressure  of  1 10  volts  against  a  resistance  of  1 1  ohms  will  give 
a  10  ampere  current.  We  now  place  the  arc-light  in  the  circuit.  What 
voltage  is  available  for  it?  Since  E  =  IR,  and  the  I  of  the  rheostat  is  10 
amperes,  while  the  R  is  5.5  ohms,  the  pressure  employed  in  sending  the  cur- 
rent through  the  tinned  iron  strip  is  55  volts.  As  we  have  a  Z>.  /*,  of  1 10 
volts  between  the  direct  line  terminals,  we  have  no — 55  or  5 5  volts  for  the 
arc-light;  and  a  current  of  10  amperes  under  this  pressure  is,  as  we  have 
seen,  more  than  enough  for  it.  By  varying  the  rheostat  resistance  we  are 
able  to  vary  the  E  and  I  to  meet  the  requirements  of  the  arc.  — 

A  practical  objection  to  the  use  of  the  Nichols  rheostat,  as  a  distributing 
agent,  is  that  it  wastes  current.  Suppose  that  we  have  at  the  direct  line 
terminals  a  current  of  no  volts  15  amperes.  The  resistance  in  the  wires 
leading  to  the  dynamo  is  then  7.3  ohms.  To  deliver  a  5  ampere  current  at 
the  apparatus  table,  we  put  a  six-sheet  rheostat  in  the  circuit :  the  total 
resistance  of  about  19.5  ohms  gives  us  a  current  of  between  5  and  6  am- 
peres. To  get  this  current,  however,  we  have  drawn  all  the  available  amper- 
age. The  objection  does  not  apply  to  the  use  of  lamp  batteries,  next  to  be 
described. 

(2)  The  lamp  battery,  suggested  by  A.  Wright  and  devised  by  E.  W. 
Scripture,  employs  the  same  principle  of  distribution  as  the  Nichols  rheo- 
stat. Two  lamps  are  connected  in  series  with  the  1 1  o  volt  direct  current 
line:  a  lamp,  say,  of  no  volts  i  ampere,  an  ordinary  32  c.  p.  incandescent 
lamp  ;  and  another,  of  the  same  or  of  higher  amperage,  but  of  lower  voltage, 
— say,  a  10  volt  i  ampere  lamp.  The  combined  resistance  of  the  two  lamps 
corresponds  to  the  total  resistance  of  the  tinned  iron  strip  of  the  rheostat. 
The  resistance  of  the  smaller  lamp  corresponds  to  the  portion  of  the  tinned 
iron  strip  included  between  the  wires  of  the  branch  circuit  leading  to  the 
apparatus.     The  wires  leading  to  the  apparatus  from  the  lamp  battery  are 


§23-    The  Electric  Citn^ent  and  Units  of  Measurement   131 


accordingly  connected  in   parallel    with    the    terminals  of  the  low  voltage 
lamp. 

The  larger  lamp  has  a  resistance  of  1 10  ohms,  the  smaller  a  resistance  of 
10  ohms.     A   pressure   of    no  volts  against  a  resistance  of  120  ohms  will 


Lamp 


To  apparatus 


Fig.  36.     Schema  of  lamp  battery. 

give  a  current  of  \\%  or  0.916  ampere.  Since  the  resistance  of  the 
smaller  lamp  is  jV  of  the  total  resistance  of  the  circuit,  it  follows  that  the 
D.  P.  between  the  terminals  of  the  smaller  lamp  must  be  W  oi"  9-1^ 
volts.  The  two  lamps  thus  make  up  a  battery  which  delivers,  for  the  appa- 
ratus circuit,  a  current  of  .916    ampere    under  a  pressure  of  9.16  volts.     . 

By  varying  the  combination  of  large  and  small  lamps,  we  may  vary  the 
pressure  and  current  available  for  the  branch  circuit.  By  arranging  a  sim- 
ple switchboard  on  the  wall  beneath  the  terminals  of  the  direct  current  line, 
we  may  provide  for  the  simultaneous  use  of  as  many  separate  lamp  bat- 
teries as  the  laboratory  requires.  Care  must  be  taken  that  the  amperage  of 
the  small  lamp  is  at  least 
as  high  as  that  of  the  large: 
otherwise  the  small  lamp 
will  burn  out  when  the  ap- 
paratus circuit  is  opened. 
For  heavier  currents,  sev- 
eral of  the  large  lamps 
must  be  placed  in  paral- 
lel, or  a  single  lamp  of 
higher  c.  p.  employed.  The 
result  of  various  combina- 
tions of  lamps  should  be 
tested  out,  and  a  record 
kept  of  the  efficiency  of 
the  batteries. 

(3)  We  said  above  that 
yV  of  the  length  of  the 
tinned    iron    strip    in    a 

Nichols  rheostat  might  give  a  current  of  5  amperes  under  a  pressure  of  1 1 
volts ;  and  we  have  just  said  that  a  lamp  battery,  made  up  as  directed,  will 


Fig.  37.     Lamp  battery. 


132  The  Reaction  Experiment 

give  a  current  of  0.916  ampere  under  a  pressure  of  9.16  volts.  These  state- 
ments must  now  be  modified.  They  hold  true  only  under  the  rather  paradox- 
ical condition  that  nothing  is  being  done  with  the  appliances,  that  no  current  is 
being  drawn  by  the  apparatus  circuit.  As  soon  as  ever  we  close  the  branch 
circuit,  we  falsify  our  calculations,  because  we  necessarily  decrease  the 
resistance  of  that  part  of  the  main  circuit  with  which  it  is  in  parallel.  For 
suppose  that  the  apparatus  circuit,  which  we  place  in  parallel  with  the  smaller 
lamp  of  the  lamp  battery,  has  a  resistance  of  3  ohms.  The  current  now  goes 
by  two  paths:  through  the  10  ohm  resistance  of  the  lamp  and  through  the 
3  ohm  resistance  of  the  branch  circuit.  What  is  the  joint  resistance  ?  It 
is  found  by  adding  the  reciprocals  of  the  several  resistances  ( the  conduc- 
tivities ) ,  and  by  taking  the  reciprocal  of  their  sum.  The  sum  of  the 
reciprocals  is  here  tV  +  ij  or  \%\  and  the  reciprocal  of  this,  or  the  joint 
resistance  offered  by  the  two  conductors,  is  2.3  ohms.  The  current  in 
the  total  circuit  is  now,  not  \\%^  but  Y^ti  or  0.98  ampere. 1  And  since 
E  =  IR,  the  tension  between  the  terminals  of  the  lamp  battery  is  0.98 
X2.3  or  2.25  volts, — this,  instead  of  the  assumed  9.16.  Or  suppose,  again, 
that  the  resistance  of  the  branch  circuit  is  20  ohms.  The  joint  resistance 
of  this  and  of  the  smaller  lamp  is  6.66  ohms.  The  current  in  the  total  cir- 
cuit is  therefore  0.94  ampere ;  2  the  tension  between  the  terminals  of  the 
lamp  battery  is  6.26  volts.  And  what  holds  of  the  apparatus  circuit  in  par- 
allel with  the  smaller  lamp  of  the  lamp  battery,  holds  also  of  the  apparatus 
circuit  in  parallel  with  a  small  length  of  tinned  iron  strip  in  the  Nichols 
rheostat.  The  closure  of  an  apparatus  circuit  must  always  lessen  the  resis- 
tance of  the  lamp  or  strip  with  which  it  is  in  parallel,  and  must  therefore 
alter  the  current  strength  of  the  total  circuit  and  the  pressure  between  the 
terminals  of  that  part  of  it  which  is  in  parallel  with  the  branch  circuit.  The 
higher  the  resistance  of  the  apparatus  circuit,  the  less,  of  course,  does  this 
interference  with  the  primary  circuit  become. 

The  Measurement  of  Current^  Pressure  and  Resistance. — The 
preceding  Section  has  made  it  clear  that  we  cannot  work  intelli- 
gently with  electrical  appliances  in  the  laboratory  unless  we  are 
able  numerically  to  determine  the  I,  E  and  R  of  our  active  cir- 
cuits. Instruments  have  been  devised  which  make  such  deter- 
minations an  easy  matter.  Current  strength  is  measured  by  the 
amperemeter  or  ammeter ;  pressure  by  the  voltmeter;  and  resis- 
tance by  the  substitution  of  numerical  values  (gained  by  help  of 
the  ammeter  and  voltmeter)  for  I  and  E  in  the  equation  R  = 

E 

— .     The  ammeter  and  voltmeter  belong,  both  alike,  to  the  class 

1  Of  which  \\,  or  0.75  ampere,  is  flowing  through  the  apparatus  circuit. 
*  Of  which  \%^  or  0.31  ampere,  is  flowing  through  the  apparatus  circuit. 


§  23-    The  Electric  Current  and  Units  of  Measurement   133 

of  instruments  known  as  galvanometers,  which  measure  the 
strength  of  an  electric  current  by  means  of  its  electromagnetic 
action.  Very  many  forms,  both  of  ammeter  and  voltmeter,  are 
on  the  market ;  the  principles  underlying  their  construction  and 
use  are  as  follows. 


(1)  Ammeters. — We  saw  above  that  current  electricity  evinces  magnetism 
tangentially  to  its  flow  ;  /.  e,,  that  a  live  wire  is  itself  magnetic  at  right  angles 
to  the  flow  of  the  current.  If,  now,  we  coil  the  wire  in  the  shape  of  a  helix, 
we  multiply  its  tangential  action^  and  thus  obtain  stronger  magnetic  effects 
with  a  given  current.  Such  an  arrangement  is  termed  a  solenoid,  from  the 
channelled  or  pipe-like  form  of  the  conductor.  If,  again,  we  insert  in  the 
helix  a  bar  of  soft  iron,  the  effect  of  the  current  passing  along  the  wire,  in- 
stead of  diffusing  through  space,  is  concentrated  upon  the  iron  core,  and  the 
magnetism  set  up  is  still  further  intensified. 

In  the  ammeter  shown  in  Fig.  38,  ^  is  a  curved  solenoid  wound  with 
coarse  wire,  and  containing  a  hollow  soft 
iron  core.  When  a  current  is  sent  through 
the  wire,  the  solenoid  becomes  magnetic, 
and  tends  to  suck  into  its  interior  any  small 
piece  of  iron  placed  near  it,  with  a  force 
directly  proportional  to  the  strength  of  the 
current.  This  action  is  exerted  upon  the 
scythe-shaped  piece  of  thin  iron,  B,  which  is 
accurately  pivoted  at  C.  To  the  angle  of  B 
is  attached  a  light  pointer,  Z>,  travelling 
over  a  section  of  a  circular  scale.  The  unit 
of  the  scale  is  i  ampere. ^  Since  the  wire 
of  the  solenoid  is  short  and  coarse,  the  re- 
sistance offered  by  the  ammeter  is  negli- 
gible as  compared  with  the  resistance  of  the 
circuit  into  which  it  is  introduced.  All  that 
we  have  to  do,  then,  in  order  to  measure  the 
strength  of  current  in  a  given  circuit,  is  to 
break  this  circuit   and  insert    the  ammeter 

by  the  binding  posts  E^  E':  the  deflection  of  the  pointer  from  zero  informs 
us  directly  of  the  strength  of  current  which  we  are  using. 

(2)  Voltmeters. — Suppose  that  we  wind  the  solenoid  of  the  ammeter 
shown  in  Fig.  38  with  a  very  long,  fine  wire.  We  shall  then  have  reversed 
the  conditions  of  its  use.     So  far  from  the  resistance  of  the  ammeter  being 


Fig.  38. 
Ammeter.      From    S.  Bottone, 
Electricity    and     Magnetism, 
1893.  153- 


1  The  ammeter  is  graded  either  by  means  of  a  constant  battery  and  decomposi- 
tion cell,  or  by  comparison  with  another,  standard  ammeter.  Details  will  be  found 
in  the  handbooks  of  electricity  and  magnetism. 


134 


The  Reaction  Experiment 


negligible  as  compared  with  the  resistance  of  the  circuit  into  which  it  is  in- 
troduced, the  resistance  of  this  circuit  will  itself  be  negligible  as  compared 
with    the  resistance   of   the    ammeter.     Let  us  now  connect  such  a  high- 


FiG.  39. 
Simple  forms  of  ammeter  and  voltmeter,  for  ordinary  laboratory  use. 


resistance  ammeter  in  parallel  (on  a  shunt  circuit)  with  the  circuit  whose 
voltage  we  wish  to  determine.  Very  little  current  will  pass  through  the 
ammeter  wire;  the  voltage  of  the  original  circuit  will,  therefore,  remain 
practically  unchanged.  Since,  however,  the  resistance  of  the  ammeter  is 
fixed,  what  little  current  is  drawn  by  it  will,  by  Ohm's  Law,  be  proportional 
to  the  voltage  of  the  circuit.  Hence,  if  the  ammeter  is  provided  with  a 
suitable  scale,  we  may  read  off  from  it,  directly,  the  D.  P.  between  the 
points  of  the  original  circuit  at  which  it  was  inserted.  The  ammeter  has 
thus  been  transformed  into  a  voltmeter. 

Let  us  say,  e.  ^.,  that  the  solenoid  has  a  resistance  of  500  ohms.  A  pres- 
sure of  I  volt  against  this  resistance  would  give  a  current  of  ^\-q  am- 
pere; a  pressure  of  2  volts,  a  current  of  ^J^  ampere;  a  pressure  of  100 
volts,  a  current  of  \  ampere.  Let  the  unit  of  the  scale  be  a  deflection  of 
the  pointer  that  corresponds  to  the  passage  of  a  ^5^^  ^  ampere  current,  and 
let  the  scale  be  composed  of  100  such  units.  Then  we  have  a  voltmeter 
that  ranges  between  o  and  100  volts.  If,  for  example,  the  pointer  of  the 
instrument,  in  a  given  case,  stands  at  21.5,  this  means  that  the  D.  P.  be- 
tween the  points  of  the  main  circuit  which  the  shunt  circuit  connects  is  21.5 
volts. 

It  is  customary  to  allow  a  resistance  of  at  least  50  ohms,  in  the  voltmeter, 
for  every  volt  to  be  measured.  That  a  branch  circuit  of  high  resistance, 
placed  in  parallel  with  a  main  circuit,  does  not  appreciably  alter  the  D.  P. 
between  the  points  of  the  main  circuit  at  which  it  is  attached,  has  been 
sufficiently  explained  in  our  account  of  the  lamp  battery.  Hence  we  need 
not  fear  that  the  voltmeter,  any  more  than  the  ammeter,  will  sensibly  alter 
the  circuit  into  which  it  is  introduced. 

(3)  Measurement  of  Resistance. — The  arrangement  for  measuring  re- 
sistance by  help  of  ammeter  and  voltmeter  is  shown  in  Fig.  40,  where  L  is 


§  23.  The  Electric  Current  and  Units  of  Measurement  135 
a  lamp  of  unknown  resistance,  A  the  ammeter,  and  V  the  voltmeter.  We 
know,  from  Ohm's  Law,  that  R=y-     If?  then,  the  ammeter  registers  0.5 

and  the  voltmeter  109.5,  we  know  that  the  resistance  of  the  lamp  is  ^%%^i 
or  219  ohms. — 


51 


loJ 


Fig.  40. 

There  are  various  methods  of  measuring  resistance  in  a  more  direct  way. 
The  instrument  most  commonly  employed  is  some  form  of  the  Wheatstone 
bridge,  a  schematic  representation  of  which  is  given  in  Fig.  41.     When  the 


Fig.  41. 
Wheatstone  bridge.     Cf.  S.  P.  Thompson,  Elementary  Lessons  in  Electricity  and 
Magnetism,  1904,  416. 


current  which  starts  from  C  has  reached.  /*,  the  potential  will  have  fallen  a 
certain  value.  The  current  now  divides  :  the  potential  in  the  upper  branch 
.showing  a  fall  at  M^  and  a  further  fall  at  g,  and  that  in  the  lower  branch 


136 


The  Reaction  Experiment 


falling  similarly  at  TV^and  Q.  Now  if  iV  be  the  same  proportionate  distance 
along  the  resistances  PNQ  that  M  is  along  the  resistances  PMQ^  the  fall 
of  potential  at  N  will  be  identical  with  the  fall  at  M;  or,  in  other  words,  if 
A:  C=B:  D,  M  and  TV  will  be  at  equal  potentials.  A  galvanometer 
placed  between  M  and  N  will  then  show  no  deflection  of  its  needle  as  the 
current  passes.  If,  therefore,  we  have  known  and  variable  resistances  at 
A^  B,  C,  and  an  unknown  resistance  at  D,  we  have  only  to  adjust  the  bridge 
so  that  the  galvanometer  needle  is  not  deflected  by  the  current,  and  we  can. 

calculate  D  from  the  formula  D=  —' 

A 

A  simple  form  of  bridge,  useful  for  laboratory  purposes,  is  the  metre 

bridge,  shown  in  schema  in  Fig.  42.     A  thin  wire  ab^   i  m.  in  length,  is 


Fig.  42. 
Metre  bridge.     From  S.  P.  Thompson,   Elementary   Lessons  in   Electricity  and 
Magnetism,  1904,  420. 

Stretched  between  stout  pieces  of  copper.  A^  B^  C,  D  are  resistances  con- 
nected similarly  to  copper  pieces.  If  the  galvanometer  wire  is  slid  along 
ab  to  the  point  at  which  the  needle  shows  no  deflection  when  a  current 
passes,  we  \i2iVQA-\-a'.B-\-b=C'.D.  Since,  by  hypothesis,  all  resistances 
but  one  are  known,  the  unknown  resistance  may  be  calculated.  In  a  simpler 
form  of  bridge,  A  and  B  are  replaced  by  copper  strips  of  inappreciable  re- . 
sistance  ;  so  that  a\b=C'.  D.  If  C  is  a  known  and  D  an  unknown  resis- 
tance, the  ratio  of  the  lengths  a  and  b  enables  us  at  once  to  calculate  the 
ohmage  of  D. — 

It  is  advisable  that  the  resistance  of  all  apparatus  in  the  laboratory, 
which  may  be  introduced  into  an  electric  circuit,  be  determined,  and  a  tag 
showing  its  ohmage  attached  to  each  instrument. 

Dynamos  and  Motors. — We  know  that,  whenever  a  current 
flows  along  a  wire,  a  magnetic  field  is  set  up  round  about  the 
wire. ,  It  is  true,  conversely,  that  if  a  magnetic  field  is  brought 
near  a  wire  currents  are  *  induced '  in  the  wire.  It  is  upon  this 
principle  of  electromagnetic  induction  that  the  action  of  the 
dynamo  depends. 


(i)  Simple^  direct  current  dynamo.  —  Suppose  that  we  rotate  a  coil  of 


§  23.   The  Electric  Current  and  Units  of  Measurement  137 

wire  in  the  magnetic  field  between  the  poles  of  a  magnet  (permanent  magnet 
or  electromagnet).  If  the  ends  of  the  coil  are  joined,  so  that  the  wire  forms 
a  complete  circuit,  currents  will  be  induced  in  it, — currents  that  flow  first  in 
one  direction  and  then  in  the  other,  according  to  the  direction  in  which  the 
lines  of  force  of  the  magnetic  field  pass  through  the  coil.  To  convert  these 
alternate  currents  in  the  coil  or  armature  into  a  continuous  current  in  the 
external  circuit,  we  connect  the  two  ends  of  the  coil  to  the  two  halves  of  a 
split  tube  (the  commutator).  As  the  coil  revolves,  the  commutator  segments 
revolve  also.     They  are  made  to  turn  between  two  strips  of  copper,  termed 


Fig.  43. 
Schemata  of  direct  current  and  alternating  current  dynamos. 
Glazebrook,  Electricity  and  Magnetism,  1903,  392. 


From   R.  T. 


the  brushes,  which  are  in  connection  with  the  external  circuit.  It  is  clear 
that,  at  every  half  turn  of  the  armature,  its  connection  with  the  external 
circuit  is  reversed  in  direction  by  means  of  commutator  and  brushes  ;  and 
this  means  that  a  continuous  current  flows  in  the  outside  circuit.  The 
essential  parts  of  the  simple  dynamo  are,  then,  the  field  magnet,  the  armature, 
and  the  commutator  and  brushes. 

(2)  Alternating  cut-rent  dynamo. — In  this  form  of  dynamo,  the  split  ring 
commutator  is  replaced  by  two  slip  rings  (collector  rings),  to  which  the  ends 
of  the  armature  wire  are  attached.  The  current  in  the  external  circuit  now 
changes  in  direction  with  the  changes  of  current  in  the  armature. 

(3)  Excitation  of  field  coils. — The  field  magnet  of  a  dynamo  is  usually 
an  electromagnet.  In  the  alternating  current  machine,  the  coils  of  the  magnet 
must  be  supplied  with  a  separate  starting  current.  This  is,  however,  not 
necessarily  the  case  with  the  direct  current  dynamo.  For  suppose  that 
the  field  coils  are  connected  with  the  armature,  through  the  brushes,  and 
that  the  machine  is  started.  There  will  be  enough  residual  magnetism  in 
the  iron  of  the  machine  to  set  up  a  small  current :  this  current,  passing  round 
the  field  coils,  magnetises  them  more  strongly  :  the  stronger  magnetisation 
of  the  coils  increases  the  current,  which  again  reacts  on  the  field  :  and  so 
on.     The  dynamo  is  then  self-excited. 

(4)  Series  and  shunt  winding.  —  The  field  coils  of  the  direct  current 
dynamo   may  be  connected  with    the  brushes  and  thus  with  the  external 


138 


TJie  Reaction  Experiment 


'Circuit  in  two  ways  :  in  series  and  in  parallel.  In  the  former  case,  the  whole 
•current  flows  round  the  field  coils,  which  therefore  consist  of  a  few  turns  of 
thick  wire.  In  the  latter  case,  the  field  coils  consist  of  a  large  number  of  turns 
of  thin  wire.  Series  wound  dynamos  are  used  where  a  constant  current  of 
varying  pressure  is  required  {e.  g.^  for  arc  lighting) ;  shunt  wound  dynamos 
where  a  varying  current  at  constant  pressure  is  demanded  (^.  ^.,  forincandes- 


FiG.  44. 

Schemata   of    series-wound  and  shunt-wound  dynamos.       Cf.   S.  P.  Thompson, 

Elementary  Lessons  in  Electricity  and  Magnetism,  1904,  483. 

cent  lighting).  Shunt  winding  is  often  replaced  by  compound  winding, 
where  some  of  the  coils  are  in  series  with  the  external  circuit,  but  in  addi- 
tion a  number  of  turns  of  thin  wire  are  wound  on  in  parallel  with  the  exter- 
nal circuit. 

(5)  Direct  current  motors.  —  In  the  dynamo,  mechanical  is  transformed 
into  electrical  energy.  In  all  direct  current  machines,  this  transformation  is 
reversible  :  by  supplying  a  current  to  the  machine,  we  cause  the  armature 
to  rotate,  and  thus  transform  electrical  energy  into  mechanical.  We  then 
have  an  electromotor.  It  may  be  worth  while  to  remember  that  a  series 
wound  motor  always  rotates  in  the  opposite  direction  to  the  series  wound 
dynamo  ;  while  a  shunt  wound  motor  rotates  in  the  same  direction  as  the 
shunt  wound  dynamo,  or  in  the  opposite  direction,  according  to  its  mode  of 
connection  with  the  source  of  supply.i  A  motor  should  be  started  up  grad- 
ually, by  help  of  an  adjustable  resistance  in  the  external  circuit. 

^  A  shunt  wound  motor,  in  which  both  field  and  armature  are  connected  to  a 
source  of  like  polarity  (-+-  to  +  and —  to  — ),  will  rotate  in  the  same  direction  as 
the  dynamo.  To  reverse  the  direction  of  rotation,  the  polarity  of  the  source  ap- 
plied to  field  or  armature  must  be  reversed.  A  compound  wound  motor  will  ro- 
tate in  the  opposite  direction  to  the  dynamo  if  the  series  winding  is  the  more 
powerfiil,  and  will  behave  as  a  shunt  wound  motor  if  the  shunt  winding  is  the 
more  powerful. 


§  23-    The  Electric  Current  and  Units  of  Measurement   139 

A  very  great  variety  of  direct  current  motors  is  on  the  market.  In  a  lab- 
oratory supplied  with  the  direct  current,  motors  of  this  type  are  indispensa- 
ble as  colour  mixers  (vol.  i.,  S.  M.,  6).  They  may  also  be  employed  to 
drive  kymographs  (104) ;  to  rotate  the  Masson  disc  (m) ;  to  turn  the  rota- 
tion apparatus  (113);  to  move  the  pressure  or  temperature  point  of  the 
kinesimeter  (i.,  I.  M.,  93) ;  to  rotate  the  rhythm  apparatus  (349)  ;  to  revolve 
the  barrel  and  cylinder  of  the  phonograph ;  and  for  many  other  purposes. 
•  (6)  Conversions  of  the  direct  curreiit  dynamo. — It  is  evident  that,  if  we 
connect  the  commutator  segments  of  a  direct  current  dynamo  with  collector 
rings  placed  farther  out  along  the  shaft,  we  may  draw  from  it  an  alternating^ 
current.  A  machine  of  this  sort  will  produce  direct  or  alternating  current^ 
as  required  ;  it  is  termed  a  double-current  generator.  If,  now,  we  run  the 
dynamo  as  a  motor,  and  deliver  a  direct  current  to  the  direct  current 
brushes,  we  may  again  draw  off  an  alternating  current  from  the  collector 
rings  ;  direct  has  been  transformed  into  alternating  current,  and  the  machine 
is  an  inverted  converter.  If,  finally,  after  the  machine  has  attained  its  full 
speed,  we  deliver  to  the  collector  rings  alternating  currents  of  the  same 
periodicity  as  the  dynamo  itself  will  produce,  we  may  draw  off  direct  cur- 
rent from  the  direct  current  brushes  ;  alternating  has  been  transformed  into 
direct  current,  and  the  machine  is  a  rotary  converter.  The  rotary  con- 
verter is  typical  of  the  class  of  machines  known  as  synchronous  motors. 

(7)  Alternating  current  motors.  —  There  is,  however,  another  class  of 
alternating  current  motors,  the  non-synchronous  or  induction  motors,  which 
is  more  generally  employed  in  laboratory  work.  In  these  motors,  an  iron 
ring  is  supplied  with  separate  windings  of  wire,  to  which  are  delivered  alter- 
nating currents  differing  in  phase  by  a  known  and  constant  amount.  A 
magnetic  field  thus  travels  round  the  ring  with  a  constant  periodicity.  If, 
now,  a  closed  armature  coil  be  mounted  within  the  ring,  the  travelling  mag- 
netic field  induces  currents  in  the  coil ;  and  these  currents  in  their  turn 
react  upon  the  field  of  magnetic  force  and  tend  to  move  across  it.  In  other 
words,  the  armature  is  carried  round  with  the  magnetic  field,  and  we  have 
an  induction  motor.  Of  this  type  are  the  60  cycle  no  volt  fan  motors, 
which  can  be  inserted  directly  into  a  lamp  socket,  and  are  much  used  in 
places  of  business. 

Induction  motors  are  simple  in  construction,  have  no  moving  contacts, 
and  do  not  easily  get  out  of  order.  They  may  be  so  varied  as  to  possess 
the  properties  either  of  the  series  wound  or  of  the  shunt  wound  direct  cur- 
rent motor.  Hence  a  laboratory  that  is  supplied  with  the  alternating,  in- 
stead of  the  direct,  current  may  readily  procure  a  set  of  induction  motors 
suitable  to  its  requirements  ;  though,  as  a  matter  of  fact,  the  majority  of 
motors  designed  for  purely  psychological  purposes  have  been  of  the  direct 
current  type.  For  work  with  other  instruments  than  motors,  the  alternating 
must  be  transformed  into  direct  current, — e.  g.,  by  means  of  the  rotary  con- 
verter mentioned  under  (6), — which  may  then  be  distributed  through  the 
laboratory  by  the  Nichols  rheostat  or  by  lamp  batteries. 


140  TJie  Reaction  Experiment 

Practical  Hints  for  Care  of  Apparatus. — Many  of  the  small 
direct  current  motors  on  the  market  are  extremely  durable  if 
properly  cared  for,  but  soon  become  useless  if  neglected.  A 
cardinal  rule  is  that  the  motor  must  be  kept  clean  and  dry ;  it 
should  be  gone  over  with  a  cotton  cloth,  before  and  after  using, 
to  free  it  of  dust  and  caked  oil ;  and  it  should  never  be  allowed 
to  stand  uncovered  in  the  laboratory.  The  oil  used  for  lubrica- 
tion should  be  of  the  best  quahty,  and  should  be  carefully  applied. 
The  commutator,  which  is  perhaps  the  most  sensitive  part  of  the 
motor,  must  be  kept  smooth  by  rubbing  with  fine  emery  cloth  ; 
no  emery  dust  must  be  allowed  to  remain  upon  commutator, 
brushes  or  shaft.  After  cleaning,  the  commutator  should  be 
wiped  with  an  oily  rag,  not  oiled ;  care  must  be  taken  to  keep 
the  oil  away  from  the  insulation.  Before  starting,  the  armature 
should  be  turned  round  slowly  by  hand,  to  see  that  nothing  catches, 
that  there  are  no  loose  wires  adhering  to  it,  etc.  The  brushes 
must  be  carefully  adjusted,  so  that  the  motor  runs  without  spark- 
ing, which  is  bad  both  for  brushes  and  for  commutator. 

If  a  motor,  properly  connected  and  supplied,  fails  to  work,  we 
may  suspect  one  or  other  of  the  following  five  defects  :  burning 
out  of  the  armature,  due  to  a  jamming  of  the  whole  armature  or 
to  faulty  winding  of  a  coil ;  short  circuit  of  the  armature  ;  defects 
of  the  commutator  ;  short  circuit  of  the  field  magnet ;  disconnec- 
tion in  the  field  magnet.  Short  circuiting  and  burning  out  may '^  "* 
generally  be  detected  by  the  heating  of  the  affected  coil  and  the 
smell  of  charred  varnish.  Tests  may  be  made  by  disconnecting 
the  suspected  coil,  and  connecting  it  with  a  primary  battery  and 
a  current  tester  or  an  electric  bell.  When  the  seat  of  the  injury 
has  been  discovered,  the  defective  part  must  be  returned  to  the 
maker  or  handed  over  to  the  laboratory  mechanician. 

Where  alternating  current  motors  are  employed,  they  should 
Tdc  carefully  safeguarded  by  high-resistance  fuse  plugs,  and  it 
should  be  observed  that  they  start  at  once  when  the  current  is 
applied.  If  the  armature  jams  at  all,  the  coils  very  quickly  burn 
out. 

The  current  tester,  spoken  of  above,  does  good  service  in  the  testing  out 
of  long  laboratory  circuits.  It  consists  of  a  small,  low-resistance  electro- 
magnet, boxed  in  a  wooden  case  resembling  that  of  the  push-button  of  an 


§  24-    TJie  Tec/miqiie  of  the  Simple  Reaction  141 

electric  bell ;  the  armature  is  drawn  down  with  a  click  when  current  passes. 
Suppose,  e.  g.^  that  we  have  a  simple  Hipp  chronoscope  circuit,  and  that  the 
apparatus  refuses  to  work.  We  have  connections  at  the  clock,  at  the 
rheochord,  at  the  commutator,  at  the  stimulator,  at  the  key,  and  at  the  gen- 
erator temiinals.  By  aid  of  the  tester,  we  may  work  along  this  circuit,  be- 
ginning at  the  generator,  and  so  localise  the  point  of  defective  connection 
or  of  disconnection.  The  current  tester  may  be  replaced  by  a  telephone 
snapper  (Queen  pony  receiver,  d.  90)  or  by  a  pocket  compass  set  in  a  block 
with  a  few  turns  of  wire  about  it. 

The  rule  of  cleanliness,  which  we  have  laid  down  for  motors,  holds  for  all 
forms  of  apparatus  in  which  electricity  is  employed.  Clean  mercury,  clean 
binding  posts,  clean  wires, — these  are  essential.  A  few  minutes'  work  with 
sandpaper,  at  the  beginning  of  the  laboratory  hour,  may  prevent  the  waste 
of  an  afternoon. 

Practical  Exercises. — ( i )  Measure  the  resistance  in  the  two 
magnets  of  the  Hipp  chronoscope. 

(2)  Measure  the  resistance  in  a  given  reaction  circuit,  includ- 
ing chronoscope  magnet,  rheochord,  stimulator  and  reaction  key. 

(3)  Given  a  battery  of  primary  cells  arranged  for  the  reaction 
experiment :  measure  the  E.  M.  F.  of  the  battery,  the  strength 
of  current  dehvered,  the  internal  resistance  of  the  cells  used. 

(4)  Test  a  given  lamp  battery. 

(5)  Measure  the  drop  in  volts  along  the  strip  of  a  Nichols 
rheostat. 

(6)  Measure  the  resistance  in  the  primary  and  secondary  cir- 
cuits of  an  inductorium. 

(7)  Draw  a  diagram,  showing  the  change  of  connections  re- 
quired if  a  given  dynamo  is  to  be  employed  as  a  motor.  Include 
some  form  of  resistance. 

§  24.  The  Technique  of  the  Simple  Reaction. — In  vol.  i.,  117  ff ., 
we  discussed  the  simple  reaction  from  a  purely  qualitative  point 
of  view.  Our  unit  of  measurement  was  -gL-  sec,  and  we  used  the 
reaction  times  solely  as  a  check  upon  6^'s  introspective  record. 
We  have  now  to  repeat  the  experiment  from  the  quantitative 
standpoint.  Our  measurements  are  to  be  made,  not  with  the 
vernier  chronoscope,  but  with  an  electric  chronoscope,  whose 
unit  is  Y^oT  s^c.  The  stimulus  to  reaction  will  be  given,  and 
the  reaction  movement  recorded,  by  instruments  which  make  or 


142  The  Reaction  Experiment 

break  an  electric  circuit.  The  whole  experiment  thus  becomes 
more  complicated :  we  must  know  the  errors,  constant  and  vari- 
able, of  our  apparatus,  and  must  take  them  into  account  when 
we  seek  to  determine  the  representative  value  and  the  variability 
of  the  reaction  times.  In  the  long  run,  introspection  will  come 
to  its  rights  again  ;  but  we  shall  for  the  present  direct  our  at- 


^J>-1^-||%^ 


Fig.  45. 


Fig.  46. 


tention  mainly  to  the  physical  and  psychophysical  aspects  of  the 
experiment. 

We  must  consider,  in  order,  the  chronoscope,  the  instruments 
for  its  control,  the  stimulator,  and  the  reaction  key. 

I.  The  Hipp  Chronoscope. — The  Hipp  chronoscope  is  an  in- 
strument designed  for  the  measurement  of  short  periods  of  time 
in  units  of  icr  or  -njVo  s^^-  ^^  consists  of  three,  separate  but  in- 
terconnected parts  :  the  clockwork,  the  registering  apparatus,  and 
the  electromagnetic  mechanism. 

(a)  The  Clockwork. — The  clock  is  driven  by  a  weight,  shown 
in  Figg.  45,  46.    The  cord  is  wound  upon  a  barrel,  the  movement 


§  24-    TJie   Technique  of  the   Simple  Reaction  143 

of  which  is  transmitted  by  a  series  of  gears  to  the  shaft  carrying 
the  crown  wheel,  and  to  the  balance  wheel.  The  latter  is  shown, 
behind  the  upper  dial,  in  Fig.  45.  The  crown  wheel,  which  has  100 
teeth,  faces  towards  the  front  of  the  clock  ;  it  can  readily  be  seen 
in  the  instrument  itself.  The  movement  in  the  balance  wheel  is 
regulated  by  a  straight  steel  spring,  whose  free  end  plays  between 
the  teeth  of  the  wheel.  The  spring  is  adjusted  to  make  precisely 
1000  vibrations  in  the  i  sec.  If  the  clock  is  running  evenly  and 
at  its  right  rate,  the  balance  wheel  moves  forward  one  tooth 
while  the  spring  makes  one  vibration.  The  characteristic  tone 
given  out  by  the  vibrating  spring  assures  E  that  the  clock  is 
•running  aright. 

The  clock  is  started  by  a  pull  upon  the  nearer  of  the  two 
cords  hanging  at  the  left  of  the  instrument.  The  leverage  is  so 
arranged  that  the  pull  on  the  cord  not  only  releases  the  brake, 
but  also  gives  a  push  to  the  middle  gear  of  the  clockwork  ;  hence 
the  chronoscope  very  quickly  acquires  its  full  speed.  The  mo- 
tion is  arrested  by  a  pull  upon  the  farther  of  the  two  cords. 

{b)  The  Registering  Apparatus. — The  pointer  of  the  upper 
dial  is  fixed  to  a  spindle  which  passes  through  the  (hollow)  shaft 
carrying  the  crown  wheel,  and  is  altogether  independent  of  it. 
If  the  course  of  the  spindle  is  followed  from  the  front  wall  of  the 
clock-case  through  the  body  of  the  clock,  it  will  be  seen  to  pass, 
first  of  all,  through  the  centre  of  a  second,  fixed  crown  wheel.. 
This  wheel  is  an  exact  duphcate  of  the  movable  crown  wheel, 
which  it  faces.  In  the  space  between  the  two  crown  wheels,  the 
spindle  carries  a  cross-bar,  one  end  of  which  is  so  shaped  as  to 
fit  between  the  teeth  of  the  crown  wheels.  Then,  entering  the 
hollow  shaft  which  forms  part  of  the  clockwork,  the  spindle 
emerges  at  the  back  of  the  clock-case. 

The  spindle  which  carries  the  upper  pointer  is  connected  by 
gears  (visible  in  the  instrument,  behind  the  dial  plates)  with  the 
short  spindle  that  carries  the  pointer  of  the  lower  dial.  Hence, 
whenever  it  moves,  the  lower  spindle  and  pointer  must  move 
with  it. 

It  is  clear,  now,  that  if  the  upper  spindle  is  pushed  forward, 
so  that  the  cross-bar  engages  the  teeth  of  the  fixed  crown  wheel, 
the  clock  hands  will  not  move,  even  if  the  clockwork  is  in  motion. 


144  -^^^^  Reaction  Experiment 

If,  on  the  contrary,  the  spindle  is  pulled  iDack,  so  that  the  cross- 
bar engages  the  teeth  of  the  movable  crown  wheel,  the  hands 
will  move  as  soon  as  the  clock  begins  to  run.  The  former  is  the 
normal  state  of  affairs  :  a  brass  spring,  attached  to  the  back  of 
the  upper  dial  plate,  holds  the  cross-bar  forward,  between  the 
teeth  of  the  fixed  crown  wheel.  In  other  words,  if  E  pulls  the 
nearer  cord,  and  so  starts  the  clock,  the  pointers  will  not  move. 
But  if  E^  having  pulled  the  nearer  cord,  puts  his  hand  behind 
the  clock  and  draws  the  spindle  gently  backward,  the  pointers 
will  begin  to  revolve. 

Both  dial  plates  are  graduated  in  hundredths.  A  complete 
revolution  of  the  upper  pointer  corresponds  to  a  complete  revolu- 
tion of  the  movable  crown  wheel ;  and  since  this  has  lOO  teeth, 
and  the  unit  of  movement  of  the  clock  is  i-oVo"  s^^->  ^^  pointer 
revolves  once  in  -^-^^  sec.  or  -^  sec,  and  the  unit  of  the  dial  is 
-j-(roT  ^^^-  ^^  ^^'  '^^^  lower  pointer  revolves  once  in  lo  sec.  ;  so 
that  the  unit  of  the  lower  dial  is  -^^  sec.  In  reading  the  clock, 
therefore,  the  value  of  the  upper  dial  is  simply  appended  to  the 
value  of  the  lower.  If  the  lower  pointer  stands  at  21  +,  and  the 
upper  at  85,  the  'time'  is  2185  cr. 

(c)  The  Electromagnetic  Mechanism. — To  the  back  of  the 
clock  case  are  fastened  two  electromagnets,  with  an  armature  be- 
tween them  (Fig.  46).  The  lower  magnet  is  connected  with  the 
left-hand,  the  upper  with  the  right -right  pair  of  binding  posts  on 
the  base  of  the  instrument.  The  armature  is  joined  to  a  light, 
vertical  rod,  which  plays  upon  the  upper  spindle  at  the  point 
where  it  emerges  from  the  back  of  the  clock-case.  The  position 
of  the  armature  between  the  poles  of  the  magnets  is  regulated 
by  spiral  springs,  whose  tension  is  adjustable  by  means  of  two 
eccentric  levers,  placed  to  right  and  left  of  the  magnets,  and 
moving  over  a  circular  scale. 

If,  now,  the  armature  is  poised  between  the  two  magnets,  and 
the  clock  is  started,  the  hands  will  not  move.  If,  however,  the 
two  eccentrics  are  turned  up,  so  that  the  upper  spiral  is  relaxed 
and  the  lower  spiral  tense ;  or  if  a  current  is  sent  through  the 
lower  magnet,  so  that  the  armature  is  drawn  down  ;  then  the 
hands  will  begin  to  revolve.  And  if,  on  the  contrary,  the  ec- 
centrics are  turned  down,  so  that  the  lower  spiral  is  relaxed  and 


§  24-    The   Technique  of  the   Simple  Reaction  145 

the  upper  spiral  tense  ;  or  if  a  current  is  sent  through  the  upper 
magnet,  so  that  the  armature  is  drawn  up ;  then  the  vertical  rod 
presses  forwards  against  the  upper  spindle,  the  cross-bar  is  forced 
between  the  teeth  of  the  fixed  crown  wheel,  and  the  hands  again 
stand  still.  In  virtue  of  this  twofold  regulation  of  the  position 
of  the  armature,  by  spiral  springs  and  by  the  electric  current, 
the  chronoscope  is  able  to  register  the  period  of  time  elapsing 
between  break  and  make,  between  break  and  break,  between 
make  and  break,  and  between  make  and  make,  of  an  electric  cur- 
rent.    The  arrangements  are  as  follows. 


Fig.  47. 
Arrangement  I.     (See  p.  147.)     The  chronoscope  is  shown  in  schema,  as  a  pair  of 
electromagnets  with  their  armature.     The  circuits  include  rheochord,  com- 
mutator, sound  hammer  and  reaction  key. 


(i)  Break  to  Make. — A  current  is  sent  through  the  upper  magnet,  and  the 
eccentrics  are  turned  up.  The  current  is  stronger  than  the  spring,  so  that 
the  armature  is  held  up.  If  the  clock  is  started,  the  hands  do  not  move. 
Now  the  current  is  broken.  The  spring  pulls  the  armature  down  ;  the 
upper  spindle  flies  back;  the  hands  move.  When  the  current  is  made 
again,  the  magnet  pulls  the  armature  up,  against  the  spring ;  the  upper 
spindle  is  thrust  forward  ;  the  hands  stop. 

(ii)  Break  to  Break. — A  current  is  sent  through  both  magnets  (the  bind- 
ing posts  are  connected  in  parallel  with  the  source  of  current),  and  the  ec- 

J 


146 


The  Reaction  Experiment 


Gentries  are  turned  down.  The  armature  is  held  up  by  the  spring.  If 
the  clock  is  started,  the  hands  do  not  move.  Now  the  current  in  the 
upper  magnet  is  broken.  The  current  in  the  lower  magnet  is  stronger  than 
the  spring,  and  the  armature  is  drawn  down ;  the  hands  move.  As  soon  as 
the  current  in  the  lower  magnet  is  also  broken,  the  spring  pulls  the  arma- 
ture up  ;  the  hands  stop. 

(iii)  Make  to  Break. — A  current  is  arranged  to  pass  through  the  lower 
magnet,  and  the  eccentrics  are  turned  down.  Before  the  experiment  begins, 
the  circuit  is  left  open,  so  that  the  armature  is  pulled  up  by  the  spring.  \i 
the  clock  is  started,  the  hands  do  not  move      Now  the  circuit  is  closed  \ 


K 


^=^ 


^j-^^^^j-^ 


Fig.  48. 
Arrangement  II.     (Seep.  147.) 


the  armature  is  drawn  down  ;    the  hands  move.     As  soon  as  the  current  is 
broken,  the  spring  draws  the  armature  up  again  ;  the  hands  stop. 

(iv)  Make  to  Make. — A  current  is  arranged  to  pass  through  both  mag- 
nets (connections  as  in  ii.),  and  the  eccentrics  are  turned  down.  Before  the 
experiment  begins,  the  circuits  are  left  open,  so  that  the  armature  is  pulled 
up  by  the  spring.  If  the  clock  is  started,  the  hands  do  not  move.  Now 
the  circuit  of  the  lower  magnet  is  closed  ;  the  armature  is  pulled  down  ; 
the  hands  move.  As  soon  as  the  circuit  of  the  upper  magnet  is  closed,  the 
spring  draws  the  armature  up  again  ;  the  hands  stop. 

All  four  arrangements  have  their  uses  in  the  physical  labora- 
tory. For  purposes  of  the  reaction  experiment,  we  employ  only 
i.  and  iii.  And  since  the  movement  of  reaction  must  be  a  break 
movement, — for  only  under  this  condition  can  we  time  it  with 
sufficient  accuracy, — we  are  reduced,  in  practice,  to  three  dispo- 
sitions of  our  apparatus  :    a  shunt  circuit  with  arrangement  i., 


§  24.    The  Technique  of  the  Simple  Reaction  147 

and  a  shunt  circuit  or  direct  circuit  with  arrangement  iii.     The 
procedure  in  the  three  cases  is  as  follows. 

I.  Break  to  Make  :  Shunt  Circuit. — The  wires  from  the  battery  are  led 
to  the  supply  poles  of  a  Pohl's  commutator  (Fig.  50).  With  the  one  side 
of  the  commutator  is  connected  a  high-resistanc^  circuit  ^,  containing 
chronoscope  and  rheochord  ;  with  the  other,  a  low-resistance  circuit  B. 
containing  the  stimulator  (a  sound  hammer  :  Fig.  54)  and  the  reaction  key. 
At  the  beginning  of  the  experiment,  the  hammer  a  is  open,  the  key  b  is 
closed. 

Current  is  now  flowing  in  ^.  By  the  conditions  of  arrangement  i.,  the 
hands  of  the  clock  do  not  move.  When  a  is  closed,  and  the  stimulus  to 
reaction  thereby  given,  so  much  of  the  current  leaves  A  and  flows  into  the 
circuit  of  lower  resistance  B  that  the   clock  hands  move.     When  O  reacts, 


&_yH 


Fig.  49. 
Arrangement  III.     The  voice-key  is  shown  in  schema,  without  relay. 

by  opening  b^  the  current  is  again  confined  to  A^  and  the  hands  stop. 

II.  Make  to  Break:  Direct  Circuit. — The  battery  wires  are  led  to  the 
commutator  as  before.  Chronoscope,  hammer  and  key  are  connected  in 
series  with  the  one  side  of  the  commutator. 

At  the  beginning  of  the  experiment,  a  is  open,  b  is  closed.  By  the  con- 
ditions of  arrangement  iii.,  the  hands  of  the  clock  do  not  move.  When  a  is 
closed,  current  flows  through  the  lower  magnet ;  the  clock  hands  move. 
When  b  is  opened,  by  the  movement  of  reaction,  the  current  is  broken  ;  the 
upper  spring  comes  into  play  ;  the  hands  stop. 

III.  Make  to  Break  :    Shunt  Circuit. — The  battery  wires  are  led  to  the 


148  TJie  Reaction  Experiment 

commutator  as  before.  With  the  one  side  of  the  commutator  is  connected 
a  high- resistance  circuit  A^  containing  chronoscope,  rheochord  and  key  b  j 
with  the  other  side,  a  low-resistance  circuit  B^  containing  a  stimulator 
which  breaks  (not  makes)  its  circuit  when  the  reaction  stimulus  is  given.  As 
typical  of  this  kind  of  stimulator  we  may  take  Cattell's  voice-key  (Fig,  58). 

At  the  beginning  of  the  experiment,  both  circuits  are  closed  ;  but  so 
much  of  the  current  flows  through  B  that  the  hands  of  the  clock  do  not 
move.  The  calling  of  the  stimulus-word  into  a  breaks  B ;  the  current  is 
confined  to  A^  and  the  hands  move.  When  b  is  opened,  by  the  reaction 
movement,  there  is  no  current  in  A  ;  the  upper  spring  comes  into  play  ;  the 
hands  stop. 

While  the  electromagnetic  mechanism  makes  the  chronoscope 
available  for  our  reaction  experiments,  it  also  introduces  certain 
possibilities  of  error,  which  we  must  carefully  guard  against. 

(i)  It  is  necessary  that  \h&  direction  oi  the  current  in  the  electromagnet 
be  reversed  from  test  to  test ;  otherwise  the  cores  will  become  permanently 
magnetised.  The  instrument  usually  employed  for  this  reversal  is 
Pohl's  commutator,  shown  in  Fig.  50.  When  the  rocker  is  brought  down 
towards  E^  the  currents  in  the  two  connected  circuits  flow  in  the  directions 
indicated  in  Fig.  51,  Aj  when  it  is  thrown  over,  away  from  E^  they  flow  as 
indicated  in  Fig.  51,  ^.  Better  than  the  ordinary  form  of  the  commutator, 
with  mercury  contacts,  is  a  form  in  which  the  rocker,  as  it  is  thrown  back 
and  forth,  engages  either  one  of  two  pairs  of  spring  brass  edges. 

(2)  The  clock  hands  record  the  time  elapsing  between  break  and  make, 
or  between  make  and  break,  of  an  electric  current.  But  the  making  and 
breaking  are  not  instantaneous  ;  they  take  time.  Moreover,  they  need  not  by 
any  means  take  the  same  time.  Consider  the  simplest  possible  arrangement 
of  the  chronoscope  circuit,/.  ^.,  arrangement  II.;  the  clock  hands  move  while 
the  current  is  in  the  magnet.  Now,  if  the  current  is  strong,  the  time  of 
magnetisation  (make)  will  be  short,  and  the  time  of  demagnetisation  (break) 

will  be  long ;  if  the  current  is  weak, 
the  time  of  magnetisation  will  be  long, 
the  time  of  demagnetisation  short. 
Here,  then,  is  the  error  which  we  must 
seek  to  eliminate.  We  must  take  ac- 
count of  the  latent  times  of  the  electro- 
magnetic mechanism :  of  the  time 
during  which  the  magnet  is  growing 

Pohl  commentator.  ^^^o^g  ^^^^g^  ^o  pull  down  the  arma- 

ture against  the  spring,  and  of  the  time 
during  which  it  is  becoming  weak  enough  to  allow  the  spring  to  pull  the 
armature  up.  These  times  may  be  quite  considerable,  as  compared  with  the 
unit  of  the  clock. 


§  24.    T/ie   Technique  of  the  Simple  Reaction  149 

As  a  rule,  the  times  which  the  clock  records  are  checked  or  controlled  in 
the  following  way.  The  magnet  springs  are  set  at  a  certain  tension,  and  a 
current  is  employed  of  such  strength  that  the  armature  moves  freely  and 
easily  under  its  influence.  We  then  introduce  into  the  chronoscope  circuit 
an  adjustable  resistance  (rheochord),  and  some  control  in-  ■.*— 

strument  {e.g.^  a  falling  hammer)  which  allows  a  known       ^\  y* 

interval  {e.  g.  1500-)  to  elapse  between  make  and  break 
a  current.  The  clock  is  allowed  to  record  the  times 
fall  in,  say,  ten  successive  trials.  If  the  average  time,  as 
recorded,  is  more  than  1500-,  the  current  is  weakened,  and 
the  tests  repeated  ;  until  presently  the  clock  records  1 500- 
as  the  average  value  of  the  ten  trials.  If  the  average 
time  first  recorded  is  less  than  1500",  the  current  may 
be  strengthened  (by  the  addition  of  new  cells,  etc.)  or  the 
tension  of  the  springs  may  be  varied.  Experience  has 
shown,  however,  that  it  is  advisable  to  shift  the  springs  as 
little  as  possible.  In  any  case,  the  average  time  recorded 
must  be  brought  up  to  \^oa.  The  clock  has  now  no 
constant  error;  the  slight  variation  of  the  recorded  times 
(the  MV oi  10  determinations  should  not  exceed  ii-So")  ^^'  ^^' 

enables  us  to  estimate  its  variable  error.  So  long  as  the  tension  of  the 
springs  and  the  strength  of  the  current  remain  the  same  we  can  work  with 
times  in  the  neighbourhood  of  1500-,  and  rely  upon  the  clock  to  furnish  us 
with  correct  time  values. 

Under  these  circumstances,  we  know  nothing  either  of  the  magnitude 
(absolute  or  relative)  of  the  latent  times  of  the  magnet  or  of  the  mechanical 
accuracy  of  the  clockwork.  If,  then,  we  change  to  the  longer  times  of  com- 
pound reactions,  we  must  correspondingly  change  the  height  of  fall  of  the 
hammer,  and  must  test  the  chronoscope  over  again. — 

It  is  clear  that  we  have  here  presupposed  a  test  of  the  control  instrument 
itself.  The  test  is  made  by  the  graphic  method.  Thus,  a  tuning  fork,  of 
known  vibration  rate,  may  be  allowed  to  write  its  curve,  during  the  time 
of  fall,  upon  a  piece  of  smoked  paper  attached  to  the  hammer.  Or  we  may 
have  recourse  to  a  more  elaborate  set  of  instruments  (recording  chrono- 
graph), and  record  the  moments  of  make  and  break  upon  the  surface  of  a 
rotating  cylinder,  alongside  of  the  time-line  traced  by  an  electrically  driven 
tuning-fork. 

The  strength  of  the  current  in  the  chronoscope  circuit  is  regulated,  from 
day  to  day,  by  aid  of  the  adjustable  resistance  and  of  some  form  of 
galvanometer. 

It  need  hardly  be  said  that  what  holds  of  arrangement  II. 
holds  also,  mutatis  mutandis^  for  arrangements  I.  and  III. 

Gene7'al  Rules  for  the  Use  of  the  Chronoscope. — The  chrono- 
scope is  furnished  with  a  glass  bell-cover,  which  fits  over  the 


150  TJie  Reaction  Experiment 

clock-case  on  the  upper  platform.  The  cover  should  always  be 
kept  in  place,  even  during  a  series  of  experiments.  When  the 
clock  is  out  of  use,  it  should  be  entirely  covered  by  a  wooden  or 
cardboard  case.  It  is  essential  that  the  works  be  kept  free  from 
dust. 

The  clockwork  runs  only  for  i  min.  The  chronoscope  is 
wound  from  the  centre  of  the  lower  dial ;  the  winding  is  done 
overhand,  from  right  to  left.  The  newest  glass  bells  have  an 
opening,  through  which  the  key  may  be  inserted.  If  the  cover 
is  of  the  older  pattern,  it  must  be  lifted  with  the  left  hand  while 
the  clock  is  wound  by  the  right.  At  the  end  of  a  series  of  ex- 
periments, the  clockwork  should  be  allowed  to  run  down,  so 
that  the  weight  comes  to  rest  upon  the  base  of  the  instru- 
ment. 

The  clockwork  should  be  started  by  a  quick,  firm  pull  upon 
the  nearer  cord.  If  the  pull  is  hesitating  or  jerky,  the  spring 
will  give  out  a  hoarse  whirr,  instead  of  its  proper  clear  tone.  The 
whirr  may  presently  pass  over  into  the  tone,  or  the  clock  may 
have  to  be  stopped  and  restarted.  In  any  event,  a  jerky  or 
hesitating  pull  is  bad  for  the  works.  In  some  clocks,  it  is  pos- 
sible to  pull  the  cord  too  far,  and  so  to  interfere  with  the  start- 
ing. In  this  case,  a  cork  buffer  should  be  introduced,  to  limit 
the  extent  of  pull. 

In  some  instruments,  again, — the  fault  does  not  seem  to  be 
universal, — the  tone  of  the  spring  is  liable,  without  warning,  to 
drop  an  octave.  The  change  of  rate  appears  to  be  due  to  a 
loosening  of  the  screws  by  which  the  regulating  spring  is  ad- 
justed. E  must  be  on  the  lookout  for  this  source  of  error  ;  when 
once  its  presence  has  been  detected,  the  tone  is  easily  controlled 
by  ear. 

We  said  above  that,  if  the  lower  pointer  stands  at  21 +,  and 
the  upper  at  85,  the  *time'  is  21850-.  Now  when  the  upper 
pointer  is  between  80  and  100,  the  lower  pointer  (at  least  in  all 
the  clocks  that  the  author  has  seen  in  use)  stands  very  close  in- 
deed to  the  scale  mark  that  it  ought  not  to  reach  until  the  upper 
pointer  has  reached  100.  E  thus  runs  the  risk,  if  he  records  by 
first  impression,  of  misreading  the  time  of  the  clock  by  loocr;  he 
is  likely  to  write  down  2285  instead  of  2185.     It  must  be  re- 


§  24-    The   Technique  of  the  Simple  Reaction  151 

membered  that,  if  the  upper  pointer  has  travelled  to  85,  the  lower 
pointer  will  have  travelled  far  beyond  the  scale  mark  which  cor- 
responds with  the  o  of  the  upper  dial. 

Since  the  clock-times  increase  from  reaction  to  reaction,  and 
since  the  reaction  time  is  found  by  subtracting  the  time  recorded 
at  the  beginning  of  the  experiment  from  the  time  recorded  at 
the  end,  it  is  advisable  to  write  the  column  of  figures  in  the 
record-sheet  from  below  upwards.  The  lowest  figure  then  shows 
the  time  at  which  the  series  of  reactions  began,  the  next  shows 
the  time  at  which  the  first  reaction  was  completed,  the  topmost 
shows  the  time  at  which  the  last  reaction  was  completed.  The 
figures  are  in  the  right  position  for  the  successive  subtractions, 
and  mistakes  are  less  likely  to  arise  than  if  the  column  had  been 
written  from  above  downwards  in  the  ordinary  way. 

II.  Control  Instruments. — The  chronoscope,  as  we  have  seen, 
is  tested  by  the  help  of  some  instrument  which  enables  its 
pointers  to  record  a  known  and  constant  period  of  time.  The 
easiest  way  of  obtaining  such  a  time  is  to  allow  a  body  of  con- 
stant mass  to  fall  through  a  constant  distance.  Since  the  begin- 
ning and  end  of  the  time  must  be  marked  off  by  electrical 
means,  the  falling  body  must  make  or  break  electrical  contacts 
at  two  points  upon  its  course.  Three  types  of  instrument  are 
now  employed  for  the  giving  of  control  times  :  the  gravity  chro- 
nometer, the  pendulum,  and  the  hammer.  In  the  first,  a  heavy 
screen  falls  vertically,  and  without  appreciable  friction,  between 
guides  to  which  wheel  contacts  are  affixed.  The  control  time 
may  be  varied  by  varying  the  height  from  which  the  screen  drops 
and  the  distance  between  the  upper  and  lower  contacts.  In  the 
second,  a  heavy  pendulum  swings  over  an  arc  to  which  mercury 
or  wheel  contacts  have  been  attached.  The  control  time  is  va- 
ried by  varying  the  position  of  the  weight  and  counterweight  of 
the  pendulum,  the  angle  through  which  it  falls,  and  the  distance 
between  the  contacts  on  the  base  of  the  instrument.  In  the 
third,  a  bent  lever,  whose  longer  arm  (the  hammer)  is  heavily 
weighted,  turns  about  a  horizontal  axis ;  as  the  hammer-head 
descends,  a  cross-bar  on  the  shank  makes  or  breaks  wheel  or 
sliding  contacts.  The  control  time  is  varied  by  varying  the  posi- 
tion of  the   counterweight  on  the  short  arm  of  the  lever,  the 


152  The  Reaction  Experiment 

height  from  which  the  hammer  falls,  and  the  distance  between 
the  upper  and  lower  contacts. 

In  arrangement  I.  (Fig.  47  above)  the  control  instrument  is  placed  in  the 
low-resistance  circuit  B^  and  is  so  disposed  that  the  falling  body  first  makes 
and  then  breaks  a  contact.  While  experiments  are  being  taken,  the  cir- 
cuit is  closed  through  the  control  instrument ;  when  control  times  are  taken 
a  and  b  are  kept  closed.  In  arrangement  II.  (Fig.  48  above)  the  control 
instrument   is   introduced  into  the  chronoscope  circuit.     Experiments  and 


Fig.  52.     Simple  control  hammer. 

control  times  are  taken  as  before.  In  arrangement  III.  (Fig.  49  above) 
the  control  instrument  is  connected  with  both  circuits.  A  first  or  upper 
break  contact  is  inserted  in  B,  a  second  or  lower  break  contact  in  A. 
During  experimentation,  both  contacts  are  closed.  When  control  times  are 
taken,  a  and  b  are  kept  closed,  and  the  falling  body  is  allowed  ( i )  to  break 
the  first  or  upper  contact  in  B,  and  then  (2)  to  break  the  second  or  lower  con- 
tact in  A.  The  control  instrument  thus  functions  first  in  the  low-resistance 
and  then  in  the  high-resistance  circuit,  taking  the  place  of  a  in  the  former 
and  of  b  in  the  latter. 

Fig,  52  shows  a  simple  form  of  the  control  hammer.  The  hammer  head 
forms  the  armature  of  an  electromagnet  (actuated  by  a  battery  of  its  own) 
which  can  be  raised  or  lowered  upon  a  standard.  The  control  time  can 
thus  be  varied  by  varying  the  height  of  the  magnet ;  it  can  also  be  varied 
by  varying  the  position  of  the  counterweight,  and  the  height  of  the  upper 
contact.  A  cross-bar  on  the  shank  strikes  the  projecting  tongue  of  the  up- 
per contact ;  another  on  the  hammer  head  strikes  that  of  the  lower.  The 
hammer  falls  upon  a  rubber  cushion,  and  is  prevented  from  rebounding  by 
a  spring  catch.  The  contacts  can  be  set  either  for  make  or  for  break.  In 
arrangements  I.  and  IL,  the  upper  is -a  make,  the  lower  a  break  contact, 
and  the  two  are  connected  by  a  wire  which  permits  the  current  to  flow  when 


§  24-    The   Technique  of  the  Simple  Reaction  155 

the  instrument  is  not  in  use.  In  arrangement  III.,  both  are  break  con- 
tacts, and  the  connecting  wire  is  removed.  The  upper  limit  of  height  of 
fall,  with  this  instrument,  is  25  cm. 


III.   Stimtclators. —  (i)   For  simple  reactions  to  noise  we  may 
employ  the  instrument  shown  in  Fig.  5  3. 

A  IS  2.  spring  forceps,  of  a  size  to  hold,  with  jaws  parallel,  an  ivory  or 
steel  ball  of  2.5  cm. 
diameter.  To  the  low- 
er edge  of  the  one  jaw 
is  hinged  a  plate,  /, 
which  can  be  turned 
up  to  the  horizontal  po- 
sition, and  then  forms 
a  floor  for  the  ball  to 
rest  upon. 

To  set  the  ball  in 
place,  the  handles  are 
lightly  squeezed  with 
the  right  hand,  and 
the  jaws  opened.  Ball 
and  plate  are  brought 
up  between  the  thumb 
and  fingers  of  the  left 
hand,  until  the  plate 
strikes  against  the  op- 
posite jaw,  and  the  ball 
rests  upon  the  hori- 
zontal floor.  The  right 
hand  is  then  relaxed  ;  the  ball  is  firmly  held  ;  and  the  floor,  as  the  left  hand 
is  taken  away,  swings  down  and  out,  leaving  the  ball  a  free  space  to  fall 
through.  By  means  of  p  the  ball  can  always  be  set  at  precisely  the  same 
height. 

B  consists  of  a  board,  b,  pivoted  upon  a  horizontal  axis,  and  carrying  a 
block  of  hard  rubber,  r,  upon  which  the  falling  ball  strikes.  The  front  edge 
of  b  is  faced  with  hard  rubber,  and  rests  upon  the  pins  of  two  wheel-contacts, 
the  one  of  which  is  a  break,  the  other  a  make  contact.  The  current  passes 
from  the  forward  binding  post  of  either  side  through  the  wheel-contact,  and 
thence  along  a  wire  fastened  beneath  b  to  the  corresponding  binding  post 
at  the  far  end  of  the  base.  The  instrument  may  thus  be  used  either  as  a 
make  or  as  a  break  stimulator.  The  stimulus,  whose  intensity  may  be  varied 
by  varying  the  height  of  A  upon  its  standard,  is  the  noise  caused  by  the 
striking  of  the  ball  upon  r :  the  ball  itself  is  caught  on  the  rebound  in  a 
pocket  (not  shown  in  the  Fig.).     The  movement  of  b  about  its  axis  is  regu- 


FlG.  53.     Sound  stimulator. 


154 


The  Reaction  Experiment 


lated  by  the  set-screw  s  and  by  a  block  of  hard  rubber,  faced  with  several 
thicknesses  of  felt,  set  in  the  wooden  base  immediately  below  r. 

An  instrument  frequently  used  as  a  make  stimulator  is  the 
sound  hammer  shown  in  Fig.  5  4.     The  wires  of  the  reaction  cir- 


FiG.  54.     Sound  hammer. 

cuit  are  led  to  the  binding  posts  i,  2  ;  the  posts  3,  4  of  the 
electromagnet  are  in  connection  with  a  battery  and  key.  When 
the  electromagnetic  circuit  is  closed,  the  head  of  the  hammer  is 
drawn  down  upon  the  block  ;  a  sound  is  thus  made  at  the 
moment  that  the  reaction  circuit  through  i  and  2  is  completed. 
The  sound  may  be  graduated  over  a  fairly  wide  range  of  in- 
tensity. 

(2)  The  following  arrangement  (break  stimulator)  will  serve 
for  simple  reactions  either  to  tone  or  to  noise.  We  place  in  the 
reaction  circuit  a  double  key, — a  key  that  will  break  this  circuit 
at  the  same  instant  that  it  makes  a  second,  independent  circuit. 
The  second  circuit  includes  a  battery,  and  the  transmitter  and 
receiver  of  a  telephone  ;  the  latter  is  in  the  reacting  room.  At 
the  moment  that  the  key  is  pressed,  and  the  reaction  circuit 
broken,  the  closure  of  the  telephone  circuit  makes  a  click  in  the 
receiver.     The  click  serves  as  noise  stimulus  to  reaction. 

If  an  electric  tuning  fork  is  kept  vibrating  before  the  trans- 
mitter, a  tonal  stimulus  may  be  given.  The  pressure  of  the  key 
breaks  the  reaction  circuit,  as  before ;  the  simultaneous  closing 
of  the  telephone  circuit  means  that  the  fork-tone  begins  to  sound 
in  the  receiver. 

(3)  For  simple  reactions  to  clangs,  we  may  employ  the  instru- 
ment shown  in  Fig.  55.  A  wire  is  stretched  before  a  sounding 
box,  and  is  set  vibrating  by  means  of  a  metal  pick  held  in  the 
hand.     At   the   beginning    of   the    experiment,    wire  and    pick 


§  24-    The   Technique  of  the  Simple  Reaction  155 

are  pressed  together;  the  pluck  of  the  wire  breaks  the  low- 
resistance  circuit.  Wires  of  different  cross-section  and  tension 
may  be  used. 

The    chronoscope  circuit  of  Fig.  ^^  contains,  besides  the  reaction  key,  a 
sound  hammer  and  control  hammer.     When  clang  reactions  are  to  be  taken, 


Fig.  55. 
See  G.  Martius,  Philos.  Studien,  vi.,  1891,  403. 

the  two  hammers  are  closed.  For  comparative  purposes,  noise  reactions 
may  be  taken  with  the  sound  hammer  :  the  control  hammer  is  then  closed, 
and  the  low-resistance  circuit  (wire  and  pick  circuit)  left  open.  Control 
times  for  the  noise  reactions  may  be  taken  with  the  control  hammer  in  its 
present  position :  key  and  sound  hammer  are  closed ;  the  low-resistance 
circuit  is  open ;  the  hammer  makes  the  high-resistance  circuit  as  it  falls 
past  the  upper  contact,  and  breaks  it  again  at  the  lower  contact.  Control 
times  for  the  clang  reactions  are  taken  by  connecting  the  wires  of  the  low- 
resistance  circuit  to  an  upper  break-contact,  and  the  wires  of  the  high- 
resistance  circuit  to  a  lower  break-contact  on  the  control  hammer  :  key,  sound 
hammer,  and  wire-and-pick  are  closed.  As  the  hammer  falls,  it  first  breaks 
the  low-resistance  circuit  (the  clock  hands  move),  and  then  breaks  the  high- 
resistance  circuit  (the  clock  hands  stop). 


156 


Tlie  Reaction  Experiment 


(4)  Light  stimuli  are  usually  given  by  means  of  a  pendulum, 
a  falling  screen,  or  a  shutter. 

la)  A  pendulum  stimulator  (make)  is  shown,  in  schema,  in  Fig.  ^d.     A 

heavy  pendulum  carries  a  black 
screen,  in  which  is  a  vertical  slit, 
adjustable  by  sliding  plates.  The 
screen,  in  its  turn,  carries  a  platinum 
wire,  whose  curvature  follows  the 
swing  of  the  pendulum.  As  the  pen- 
dulum falls,  from  left  to  right,  the 
wire  passes  through  two  insulated 
pools  of  mercury,  each  of  which  is 
connected  with  a  binding  post.  As 
soon,  therefore,  as  the  front  edge  of 
the  wire  enters  the  second  mercury 
pool,  a  circuit  is  completed  between 
the  posts.  At  the  same  moment,  the 
slit  in  the  moving  screen  begins  to 
pass  across  a  similar  slit  in  a  fixed 
screen  placed  behind  the  pendulum. 
The  light  which  serves  as  stimulus 
is  reflected  through  the  two  slits  from 
a  plane  mirror,  set  up  at  a  suit- 
able distance  and  in  a  suitable  posi- 
tion behind  the  whole  apparatus. 
The  wire  is  long  enough  to  keep  the  circuit  closed  until  the  movement  of 
reaction  has  been  made. 

If  the  pendulum  is  to  be  used  as  a  break  stimulator,  the  mercury  cups 
are  placed  crosswise  to  the  swing  of  the  pendulum,  and  are  provided  with 
small  lateral  orifices,  between  which  a  delicate  bridge  of  mercury  is  formed. 
The  pendulum  screen  now  carries,  instead  of  the  wire,  a  plate  of  mica.  As 
the  pendulum  falls,  the  plate  severs  the  mercury  bridge,  and  the  contact  is 
broken.     The  device  is  shown  schematically  in  Fig.  ^^j.^ 

{p)  A  simple  form  of  the  falling  screen  apparatus,  or  gravity  chro- 
nometer, is  shown  (back  view)  in  Fig.  58.  For  purposes  of  experi- 
ment, the  space  between  the  lateral  pillars  in  front  may  be  filled,  e.  g., 
hy  black  cardboard,  in  which  a  horizontal  slit  has  been  cut.  A  strip 
-of  similar  cardboard  is  placed  between  the  cHps  at  the  back  of  the 
instrument;  so  that  O  fixates  a  black  slit  in  a  black  surface.  The  heavy 
screen,  which  falls   when  the   electromagnetic  circuit  is  broken,  is  faced 


Fig.  56. 

Schema  of  the  Wundt-Lange  optical 

pendulum. 


1  Here,    as  everywhere,    mercury  has  its   disadvantages,   and  the  wet  contacts 
may  well  be  replaced  by  wheel  contacts  of  the  kind  shown  above,  Fig.  53. 


§  24-    The   Technique  of  the  Simple  Reaction 


57 


Fig.  57.     Krille's  interrupter. 


■with  white  paper.  At  the  moment  that  the  white  begins  to  show  in  the  sHt, 
the  screen  makes  or  breaks  a  wheel  contact  attached  to  one  of  the  pillars : 
the  contacts  are  not  shown  in  the 
Fig.  Electromagnet  and  chps  are 
both  adjustable  in  the  vertical  di- 
rection. The  noise  made  by  the 
drop  of  the  screen  upon  the  base  of 
the  instrument  comes  later  than  the 
reaction  movement. 

The  gravity  chronometer  has  been 
built  in  various  forms,  and  used 
for  a  variety  of  purposes.  If  great 
accuracy  is  required,  the  pillars  may 
be  made  higher,  and  the  screen  al- 
lowed to  fall  from  a  correspondingly 
greater  height.  If  long  exposures 
are  needed,  the  principle  of  the 
Atwood  machine  may  be  applied. 
The    two  pointed    wires  projecting 

from  the  lower  edge  of  the  screen  in  Fig.  58  make  connection,  when  the 
screen  drops,  between  two  mercury  pools  ;  the  closure  of  the  circuit  coin- 
cides with  the  exposure,  through  a  slit  in  the  screen  itself,  of  a  printed 
word  or  other  visual  stimulus  on  the  card  held  by  the  clips.  This,  and  not 
the  arrangement  of  the  apparatus  for  the  simple  reaction,  is  the  arrangement 
represented  in  the  Fig. 

{c)  The  requirements  of  a  shutter  for  simple  reactions  are  that  it  shall 
work  noiselessly,  and  that  it  shall  make  or  break  a  circuit  at  the  moment 
that  exposure  begins.  These  requirements  are  most  simply  fulfilled  if  the 
shutter  be  pivoted  upon  a  horizontal  axle  which  also  carries  a  wheel  contact ; 
the  shutter  may  be  opened  by  means  of  a  weight, 

(5)  Pr^j-i-//r^  Stimulators  (area!)  for  break  and  make  are  shown 
in  Figg.  60,  61.  In  both,  the  stimulus  is  given  by  a  rounded 
hard-rubber  surface.  The  hook  at  the  tip  of  the  make  stimulator  is 
made  of  bone.  The  distance  between  the  platinum  point  on  the 
spring  and  a  corresponding  platinum  plate  on  the  bar  (not  shown 
in  Fig.  61)  is  0.5  mm.  It  is  clear  that  a  short  interval  must 
elapse  between  the  giving  of  the  pressure  stimulus  and  the  mak- 
ing of  the  reaction  circuit.  With  practice,  this  interval  becomes 
constant,  and  does  not  exceed  3cr. 


For  punctual   stimulation,  the  springs  should  be  made  very  flexible,  and 
the  hard-rubber  bosses  be  replaced  by  stout  hairs. 


158 


The  Reaction  Experiment 


Fig.  62  shows  an  arrangement  for  simple  reaction  to  electrical  stimuli 
(break  shocks  of  an  induced  current),  the  stimulus  being  applied  to  the 
reacting  finger.     A  is  an  Ewald  chronoscope,  an    electromagnetic  counter 


Fig.  58. 
See  J.  McK.  Cattail,  Philos.  Stud.,iii.,  1886,  315. 


carrying  a  disc  graduated  in  hundredths,  and  actuated  by  a  100  vs.  elec- 
tric fork.  B  is  the  reaction  key  :  the  hardrubber  button,  upon  which  the 
finger  rests,   is  pierced  by   two  fine  wires,  ending   above   in   small  plates, 


24.    The   Technique  of  the  Simple  Reaction  159 


which  are  connected  (by  way  of  binding  posts  not  shown  in  the   Fig.)  with 
the  secondary  coil   of  an  inductorium.     C  is  an  electric  bell,  arranged  to 
give  a  single  short  stroke   as  warning  signal.     Z>  is    a  rocking  key ;  and  E 
is  a  cardboard  disc  fitted  to  the  lower  surface  of  a 
kymograph  drum. 

The  drum  is  so  placed  that  the  periphery  of  the 
cardboard  disc  bears  upon  the  left-hand  lever  of  the 
rocker.  As  the  disc  moves  forward,  the  lever  drops 
into  the  depression  ab  j  at  the  moment  of  passing  a^ 
it  assumes  the  position  indicated  in  the  Fig.  The  cur- 
rent can  now  flow  through  bell  and  inductorium :  the 
signal  sounds,  and  the  make-shock  is  given.  This 
shock  is,  however,  not  felt,  since  it  is  only  after  hear- 
ing the  bell  that  O  lays  his  finger  upon  B.  The  drum 
moves  on ;  and  at  b  the  current  in  the  inductorium  is 
broken.  O  now  receives  the  stimulus  of  the  break- 
shock.  At  the  same  moment,  the  current  through 
chronoscope,  fork  and  key  is  made.  The  counter 
begins  to  record  the  vibrations  of  the  fork,  and  con- 
tinues to  do  so  until  the  circuit  is  broken  by  (9's  re- 
action. The  time  of  reaction  is  then  read  off  from 
the  chronoscope  disc,  in  hundreths  of  a  sec,  and  the 
pointer  is  brought  back  to  o  by  a  turn  of  the  wheel 
indicated  in  the  Fig.  Meantime  the  drum  is  revol- 
ving ;  at  a'  the  signal  and  make-shock  recur,  at  b'  the 
break-shock  is  given  and  the  clock  circuit  made ; 
and  so  on. — The  arrangement  may  be  simplified  by 
omitting  the  bell  signal  and  kymograph,  and  moving 
the  rocker  by  hand. 


Fig.  59. 
Jastrow's  shutter. 


(6)  A  make  stimulator  (areal)  for  temperature  is  shown  in  Fig. 
63.       The  rounded  cup  of  metal  at  the  tip,  through  which  the 


Fig.  60. 
Scripture's  touch  key. 


water  flows  and  by  which  the  stimulus  is  applied,  has  a  diameter 
of  10  mm.  and  a  height  of  13  mm.  The  receiving  vessel  on  the 
handle  is  14  mm.  in  diameter  and  28  mm.  high  ;  it  is  closed  by  a 
cork,  through  which  is  passed  a  small  thermometer  reading  to 


i6o 


The  Reaction  Experiment 


0.2°.  The  receiving  tube  is  supplied  from  a  stationary  vessel  of 
hot  or  cold  water,  also  furnished  with  a  thermometer ;  this  ves- 
sel should  stand  fairly  high,  in  order  that  the  water  may  circulate 
freely  through  the  stimulus  cup.  The  error  of  the  apparatus  is 
said  to  be  negligible. 


aiijjjjsiijjajjiijijijiaij^^ 


Fig.  61. 
Dessoir's  sensibilometer. 


By  inverting  the  position  of  the  two  arms,  on  the  pattern  of 
Fig.  60,  the  instrument  may  be  changed  to  a  break  stimulator. 


Fig.  62. 
See  O.  Dumreicher,  Zur  Messung  der  Reactionszeit,  1889,  Fig.  2. 


All  temperature  stimulators,  except  those  that  employ  the  principle  of  ra- 
diation (and  they  are  not  satisfactory),  give  a  pressure  before  they  give  the 


§  24.    The   Technique  of  the  Simple  Reaction  i6i 


required  warm  or  cold  stimulus.     O  quickly   learns,  however,  to  withhold 
the  reaction  movement  until  he  senses  the  warmth  or  cold. 


[ 


A 


Fig.  63. 
von  Vintschgau's  thermophor. 

Punctual  stimulation  may  be  effected  by  the  instrument  shown  in  Fig.  64. 
A  hollow  tin  cone,  brought  below  to  a  fine  point,  is  closed  by  a  cork,  through 
which  pass  two  glass  tubes  and  a  thermometer. 
The  tubes  are  connected  by  rubber  tubing  with 
tubulated  bottles,  the  one  of  which  contains  hot, 
the  other  cold  water.  The  temperature  of  the 
water  in  the  stimulator  may  easily  be  regulated 
by  raising  or  lowering  the  one  or  other  of  the 
bottles.  To  avoid  the  effects  of  radiation,  the 
cone  is  covered,  except  at  its  extremity,  by  a 
coat  of  gutta-percha.  (9's  hand  and  arm  rest 
in  a  plaster  mould,  and  the  skin  is  covered  with 
a  thin  sheet  of  gutta-percha,  so  pierced  as  to 
allow  of  the  stimulation  of  a  marked  tempera- 
ture spot.  The  stimulator  is  fixed  to  a  stan- 
dard, and  can  be  raised  or  lowered  by  crank  and 
gears.  Arrangements  are  made  whereby  a  cir- 
cuit is  made  or  broken  as  nearly  as  possible  at 
the  moment  of  application  of  the  temperature 
stimulus  to  the  skin. 

(7)  It  is  difficult  to  find  a  way  of  giving 
olfactory  stimuli  that  shall  be  free  of  error. 
Fig.  65  shows  a  break  stimulator.  An  oval 
capsule  of  hard  rubber  contains  the  odor- 
ous substance.  A  glass  tubule  at  the 
lower  pole  leads  by  way  of  rubber  tubing 
to  a  double  rubber  bulb,  of  the  sort  used  with  atomisers.    Two  glass 

K 


Fig.  64. 

Kiesow's  temperature 

stimulator. 


1 62 


The  Reaction  Experiment 


Fig.  65. 
Moldenhauer's  smell  stimulator. 


tubules  are  introduced  symmetrically  at  the  upper  pole  :  the  one 
leads,  by  way  of  rubber  tubing,  to  a  glass  bulb  for  insertion  into 

\^^  the  nostril ;  the  other,  by  an 

^^T^  equal  length  of  tubing,  to  a 

il  \^  break  contact.      This  latter 

consists  of  a  plate  of  alumin- 
ium, furnished  with  a  plati- 
num plate  which  rests  upon 
an  adjustable  platinum  point : 
the  mechanism  will  be-  un- 
derstood from  the  Fig.  The 
stimulator  is  clamped  to  a 
standard.  When  the  appa- 
ratus is  ready  for  use,  the 
rubber  bulb  is  squeezed,  and 
the  current  of  scented  air 
passes  out  through  the  two 
upper  tubules,  entering  the 
nose  at  the  same  moment 
that  it  throws  back  the  aluminium  plate  and  so  breaks  the  reac- 
tion circuit. 

The  inconveniences  of  the  arrangement  are  as  follows,  {a)  Separate  cap- 
sules must  be  provided  for  different  scents,  {b)  There  is  no  way  of  grading 
the  intensity  of  the  stimulus  with  any  degree  of  accuracy.  (<:)  The  noise 
from  the  rubber  bulb  is  distracting,  as  is  {d)  the  rush  of  air  into  the  nostril, 
which  gives  a  pressure  stimulus  before  the  odour  is  smelled.  {e)  The  smell 
of  the  stimulus  is  complicated  by  the  smell  of  the  rubber  tubing.  (/)  The 
reaction  circuit  is  broken  before  (and  we  do  not  know  how  long  before)  the 
odorous  stimulus  reaches  the  olfactory  mucous  membrane. 

A  make  stimulator  is  shown  in  Fig.  66.  A  hollow  handle  of 
hard  rubber  carries  a  small  metal  cup,  C,  Handle  and  cup  are 
pierced  by  the  hard  rubber  rod,  ABy  which  terminates  above  in 
a  metal  knob,  fitted  to  close  the  aperture  of  C.  A  spiral  spring, 
wound  around  AB,  tends  to  press  it  downwards,  and  so  to  open 
C;  this  action  is  prevented  by  the  pin  P.  Two  arms,  D,  at- 
tached to  ABy  carry  platinum  points ;  and  two  metal  strips,  Ey 
carrying  platinum  plates,  are  connected  with  binding  posts  at  the 
base  of  the  handle.     At  the  beginning  of  an  experiment,  the 


§24.    The   Technique  of  the  Simple  Reaction  163 


odorous  substance  (dropped  on  sponge  or  cotton  wool)  is  placed 
A  m    C.     O  withdraws  the  pin  P ;  AB  is  driven 

down  by  the  spring  ;  and  contact  is  made  between 
D  and  E  at  the  same  time  that  the  smell  stimu- 
lus is  presented.  For  a  second  experiment,  B  is 
pushed  up  until  P  catches  the  shoulder  of  the 
rod  and  C  is  closed. 


It  is  clear  that  this  stimulator  is  open  to  much  the 
same  criticism  as  has  been  passed  upon  the  other.  Better 
than  either  is  the  arrangement  shown  in  Fig.  67.  The 
flask  introduced  between  the  T-piece  of  the  inhaling  tube 
and  the  rubber  tubing  that  leads  to  the  tambour  prevents 
any  admixture  of  india  rubber  with  the  olfactometer 
stimulus.  The  writing  lever  of  the  tambour  records  on 
the  kymograph  the  whole  course  of  (9's  inspiration.  The 
reaction  key  is  arranged  for  air  transmission;  and  a 
I  co-fork  writes  a  time  line  from  which  the  duration  of 
the  reaction  may  be  read. 

(8)  Make  and  break  stimulators  for  taste  may- 
be constructed  on  the  analogy  of  Figg.  60,  61. 
The  spring  is  made  light  and  flexible,  and  the 
hard  rubber  boss  is  replaced  by  a  fine  camel' s- 
hair  brush. 


Fig.  66. 

Buccola's  smell 

stimulator. 


Fig.  67. 
Zwaardemaker's  arrangement  for  olfactory  reactions. 


1 64 


The  Reaction  Experiment 


IV.  Reaction  Keys. —  Any  form  of  voluntary  movement  — 
movement  of  hand,  foot,  lips,  jaws,  larynx,  eyelid — may  serve  as 
response  to  the  reaction  stimulus.  The  movement  most  commonly 
employed  in  work  upon  the  simple  reaction  is  a  finger  movement, 
and  the  key  is  a  finger  key  modelled  upon  the  ordinary  telegraph 


Fig.  68. 

key  (Fig.  68).  The  key  may  be  permanently  closed  (for 
control  experiments)  by  means  of  the  brass  strip  and  clutch 
attached  to  the  base.  The  forefinger  of  the  right  hand  is  laid 
upon  the  hard  rubber  cap,  thus  holding  the  key  closed  ;  the  reac- 
tion movement  consists  in  the  lifting  of  the  finger  and  consequent 
break  of  contact.  The  rest  of  the  hand  and  the  arm  are  sup- 
ported by  a  cushion,  plaster  mould,  or  what  not,  placed  con- 
veniently upon  the  table.  The  mo\'ement  may  be  a  simple  lift 
of  the  finger,  or  may  be  made  laterally  (up  and  to  the  right), 
as  O  prefers. 

If  a  downward  movement  of  the  finger  is  preferred,  the  key 
has  the  form  shown   schematically  in   Fig.  69.     The  finger  is 


r 


u 


w 


I 


a 


ezzj 


Fig.  69. 
Jastrow's  reaction  key. 

laid  lightly  upon  the  left-hand  cap,  a  sharp  pressure  upon  which 
breaks  contact.  The  key  remains  open,  after  the  reaction  move- 
ment has  been  made,  since  the  wedge-shaped  point  at  the  centre 
slips  into  the  second  notch  in  the  spring. 


§  24-     TJie   Technique  of  the  Simple  Reaction  165 

Fig.  70  shows  a  key  in  which  the  movement  of  reaction  con- 
sists in  the  opening  of  the  closed  thumb  and  forefinger. 

The  key  consists  of  two  hard-rubber  blocks  run- 
ning on  steel  guides  :  the  lower  block  may  be  fixed 
in  any  required  position  by  means  of  a  set-screw; 
the  upper  moves  up  or  down  with  the  movement 
of  reaction.  If  the  key  is  held  free  in  (9's  right 
hand,  the  forefinger  is  inserted  in  the  hole  of  the 
upper,  the  thumb  in  that  of  the  lower  block.  If 
it  is  fastened  to  a  support,  or  held  in  O's  left  hand, 
the  thumb  of  the  right  (the  reacting)  hand  is  placed 
against  the  projecting  hook  of  the  lower  block. 
The  three  binding-posts  enable  us  to  use  the  key 
for  breaking  a  circuit  by  a  movement  either  of 
flexion  (upper  and  middle)  or  of  extension  (middle 
and  lower  posts).  It  may  also  be  used,  though 
less  accurately,  as  a  make  key :  for  a  movement 
of  flexion,  the  circuit  wires  are  carried  to  the  mid- 
dle and  lower,  for  a  movement  of  extension,  to  the 
upper  and  middle  binding-posts.     The  lost  time  Fig,  70. 

of  the  make-records  may  be  reduced  to  a  minimum  Scripture-Dessoir  reaction, 
by  adjustment  of  the  lower  sliding  block.  key. 

Fig.  71  shows  a  lip  key.     The  two  ivory  plates  at  the  ends  of 


Fig.  71. 

Cattell's  lip  key. 

the  brass  arms  are  held  between  the  lips,  and  the  key  is  thus  kept 
closed  until  the  parting  of  the  lips  opens  it. 

A  speech  key  may  be  constructed  as  follows.  A  bit  of  soft 
wood,  fitted  to  the  teeth,  is  pivoted  at  the  long  end  of  a  metal 
lever,  the  short  end  of  which  plays  between  hard  rubber  and  a 
brass  contact.  The  lever  is  held  against  the  rubber  by  a  spring. 
The  wood  is  taken  between  the  teeth,  the  head  exerting  such  a 
pull  upon  it  that  the  lever  is  held  in  contact  with  the  brass.  The 
natural  movement  of  separating  the  teeth  in  speaking  frees  the 


1 66 


The  Reaction  Experiment 


short  end  of  the  lever,  which  is  thus  drawn  away  from  the  brass 
contact  by  the  spring. 


^^ 


lii::—^^ 


Jastrow's  speech  key. 


P'lG    72. 

The  spring,  which  may  be  an  ordinary  elastic  band,  is  not 
shown  in  the  Fig. 


A  voice  key  has  been  shown  in  Fig.  58.  It  consists,  first,  of 
a  funnel,  fitted  with  a  mouthpiece  at  its  smaller  and  a  screw  cap 
at  its  larger  end.  The  cap  is  covered  with  fine  leather,  carrying 
at  its  centre  (on  the  inside)  a  light  platinum  contact,  connected 
with  a  binding  post  at  the  periphery.  Across  the  ring  of  the 
cap  is  fixed  a  bar  of  metal  (also  on  the  inside),  connected  at  one 
end  with  a  binding  post  and  carrying  a  platinum  point  against 
which  the  contact  of  the  leather  rests.  When  O  speaks  into  the 
mouthpiece,  the  leather  is  driven  away  from  the  bar,  and  the 
contact  is  momentarily  broken.  To  make  this  break  permanent, 
the  circuit  passing  through  the  cap  of  the  funnel  passes  also 
through  a  relay  (a  permanent  electromagnetic  interruptor)  ;  as 
soon  as  the  platinum  contact  is  broken,  the  armature  of  the  re- 
lay magnet  is  released,  and  the  circuit  is  definitively  broken. 

Questions. —  ^  and  (9  (i)  We  have  noted  certain  sources 
of  error  in  the  use  of  Moldenhauer's  smell  stimulator.  What 
sources  of  error  are  involved  in  the  use  of  the  taste   stimulator  ? 

(2)  Suggest  a  stimulator  for  use  in  pain  reactions. 


§  24-    TJie   Three   Types  of  Simple  Reaction  167 

(3)  Work  out  the  details  of  a  temperature  stimulator  (punctual 
stimulation)  from  the  directions  given  on  p.  161. 

(4)  ''  The  Hipp  chronoscope  has  proved  itself  a  most  satis- 
factory time-piece,  and  perhaps  the  same  principle  might  be  made 
use  of  in  a  chronoscope  that  would  record  with  an  equivalent  de- 
gree of  accuracy  the  ten-thousandth  part  of  a  second  "  (Witmer, 
Psych.  Rev.,  i.,  1894,  515).  What  are  the  chief  faults  of  the 
Hipp  chronoscope  as  a  reaction  instrument }  What  is  the  signif- 
icance of  the  third  figure  of  the  recorded  times  }  What  would 
be  gained  by  recording  to  the  nearest  ten-thousandth  1 

(5)  Devise  a  portable  reaction  outfit,  that  shall  serve  for  simple 
reactions  to  light,  sound  and  pressure. 

(6)  Devise  a  pressure  stimulator  that  may  be  controlled,  like 
the  sound  hammer  or  the  light  pendulum,  by  an  electromagnet. 

(7)  To  what  various  uses  may  the  Pohl  commutator  be  put  ? 

§  25.  The  Three  Types  of  Simple  Reaction. — Our  first  task 
is  to  measure  the  time  of  the  simple  reaction,  in  its  natural,  com- 
plete and  abreviated  forms.  We  will  begin  with  reaction  to  sound  ; 
and  we  will  employ,  to  start  with,  arrangement  II  of  the  reaction 
circuit. 

EXPERIMENT   XXIV 

Materials. — Constant  current  of  50-70  milliamperes.  Gal- 
vanometer. Two-way  switch.  Hipp  chronoscope,  new  pattern. 
Rheochord.  Commutator.  Control  instrument.  Sound  hammer. 
Break  finger-key.  Wire  (no.  18  B.  &  S.  gauge). — Screen.  Cush- 
ion or  plaster  mould. 

For  magnet  of  sound  hamm,er.  Dry  battery.  Commutator. 
Wire. 

For  magnet  of  control  instrument.  Dry  battery..  Commu- 
tator.    Wire. 

For  signal  circuit.  Dry  cell.  Two  push  buttons.  Two  sig- 
nals.    Wire. 

[We  may  assume  that  the  total  resistance  of  the  clock  circuit  is 
in  the  neighbourhood  of  100  ohms.  We  shall  accordingly  re- 
quire, for  a  current  of  60  milliamperes,  a  pressure  of  6  volts. 

Most  of  the  appliances  have  been  described  above.  A  simple 
form  of  galvanometer^  supplied  by  the  makers  of  the  Hippchron- 


1 68 


The  Reaction  Experhnent 


oscope,  is  shown  in  Fig.  73.  It  has  a  range  of  o  to  100  milli- 
amperes,  and  is  accompanied  by  a  printed  table  giving  the  abso- 
lute values  of  the  deflections  of  the  needle. 

The  two-way  switch  is  shown  in  Fig.  74.     The  movable  arm 
can  be  brought  over  either  of  the  two  contact  points. 


Fig. 


ly 


Fig. 


74- 


The  rheochord  usually  supplied  with  the  accessories  to  the 
chronoscope  is  shown  in  Fig.  75.  It  consists  of  a  length  of  fine 
German  silver  wire,  looped  about  binding  posts,  and  of  two  plati- 


Fig.  75. 

num  wires,  which  pass  through  a  hard-rubber  trough  filled  with, 
mercury.  The  German  silver  wire  gives  the  coarse,  the  plati- 
num wires  give  the  fine  adjustment. 

The  screen  stands  on  (9's  table,  between  the  stimulator  and  the 
reaction  key.     The  cushion  lies  on  the  table,  before  the  key. 

The  wire  is  '  annunciator  wire,'  double  cotton  covered  and  par- 
affined. 

The  two  signals  are  actuated  by  a  single  cell  (three  wire  con- 
nection). The  signal  in  ^'s  room  may  be  an  ordinary  electric 
bell.  The  signal  in  6>'s  room  should  be  a  buzzer  or  a  single- 
stroke  bell.  The  latter  may  easily  be  made  from  the  ordinary 
bell,  by  bending  the  contact-spring  so  that  contact  is  not  broken 
when  the  current  passes  through  the  magnet.  It  is  well,  further, 
to  stick  a  lump  of  wax  to  the  inside  of  the  gong  ;  the  tone  is  thus 
rendered  shorter  and  less  resonant.] 


§  24.    The   Three  Types  of  Simple  Reaction 


169 


Disposition  of  Apparatus. — The  instruments  must  be  so  ar- 
ranged that  E  is  at  the  centre  of  the  group,  able  to  control  every 
piece  without  rising  from  his  chair.  They  must  be  so  arranged, 
also,  that  the  manipulations  take  place  in  a  natural  and  easily 
memorised  order.  Fig.  ^6  shows  a  satisfactory  disposition  of  the 
apparatus  in  arrangement  IT. 


Fig.  76. 

E  sits  on  a  swivel  chair  directly  facing  the  chronoscope.  The 
commutator  of  the  reaction  circuit  (marked  2  in  the  Fig.)  is 
placed  to  the  left  of  the  chronoscope.  Wires  running  from  the 
farther  poles  of  the  commutator  connect  in  series  the  control 
instrument,  the  sound  hammer  and  reaction  key,  the  rheochord, 
the  galvanometer  (this  only  when  the  switch  is  so  closed  as  to* 
include  the  instrument  in  the  circuit),  and  the  chronoscope. 
Rheochord  and  galvanometer  are  placed  to  the  right  of  the  chron- 
oscope, within  easy  reach  of  E ;   the  control  instrument  stands 


I/O  TJie  Reaction  Expej'iment 

on  a  separate  table  to  his  left ;  the  sound  hammer  and  key  are 
on  a  table  in  an  adjoining  room.  Commutator  3,  which  controls 
the  magnet  of  the  sound  hammer,  is  placed  to  the  right  of  the 
chronoscope ;  commutator  i,  which  controls  the  release  magnet 
•of  the  control  instrument  (pendulum,  gravity  chronometer,  or 
hammer),  is  placed  to  the  left  of  commutator  2.  The  push  but- 
ton of  the  signal  circuit  lies  to  the  extreme  right  of  E  ;  the  elec- 
tric bell  is  on  the  wall  of  the  room.  The  batteries,  indicated  in 
the  Fig.  by  the  signs  +  and  — ,  may  stand  on  or  under  the  table. 
If  batteries  are  not  employed,  the  wires  are  led  to  the  source 
of  supply,  wherever  in  the  room  that  may  be. 

The  screen  (not  shown  in  the  Fig.)  is  set  up,  on  (9's  table,  be- 
tween sound  hammer  and  key.  The  key  is  fixed  in  such  a  posi- 
tion that  (9,  seated  at  the  table  in  the  attitude  most  comfortable 
for  him,  can  lay  his  finger  easily  and  naturally  upon  the  button. 
The  cushion  or  plaster  mould  supports  his  extended  arm.  To 
the  right  of  the  key  is  a  push  button  which  rings  the  bell  in  ^'s 
room.     The  single  stroke  bell  is  on  the  wall  of  the  room. 

Course  of  an  Experiment. — We  may  anticipate  a  little,  in 
order  to  illustrate,  by  help  of  the  Fig.,  the  course  of  a  reaction 
experiment.  Let  us  suppose  that  E  and  O  are  seated  at  their 
respective  tables.  The  three  commutators  and  the  reaction  key 
are  open ;  the  control  instrument  is  closed,  so  that  current  can 
flow  directly  through  it  ;  the  galvanometer  has  been  switched 
out  of  the  reaction  circuit ;  the  clock  has  been  wound,  and  the 
time  shown  by  the  dials  has  been  recorded.  The  procedure  will 
then  be  as  follows. 

(i)  ^  places  his  left  hand  upon  the  rocker  of  commutator  2, 
and  pulls  it  down  towards  him.  The  chronoscope  circuit  is  now 
broken  only  at  the  sound  hammer  and  key.  (2)  With  the  same 
hand,  E  pulls  the  nearer  cord  of  the  chronoscope,  and  starts  the 
clockwork.  The  tone  of  the  regulating  spring  assures  him  that 
the  clock  is  running  aright.  (3)  Reaching  out  with  the  right 
hand,  he  presses  the  button  of  the  signal  circuit,  and  thus  gives 
warning  to  O  to  close  the  reaction  key.  (4)  After  a  short  2  sec, 
he  pulls  down  (with  the  same  hand)  the  rocker  of  commutator 
3,  and  thus  gives  the  reaction  stimulus.  The  clock-hands  imme- 
diately begin  to  move,  and  stop  only  on  the  breaking  of  contact 


§  24'    The   Three   Types  of  Simple  Reaction  171 

at  the  reaction  key.  (5)  As  soon  as  they  stop,  E  pulls  the  farther 
cord  of  the  chronoscope  (left  hand),  and  (6)  throws  open  the 
commutators  2  and  3  (left  and  right  hands).  Everything  is  now 
as  it  was  before  the  experiment  was  taken,  except  that  current 
has  been  passed  through  the  chronoscope  magnets  in  a  certain 
direction  and  that  the  clock-hands  have  assumed  a  different 
position.  (7)  E  then  records  the  time,  as  shown  by  the  dials. 
A  signal  from  O  (coming,  probably,  just  as  he  is  about  to  take 
the  reading  of  the  clock)  assures  him  that  affairs  have  gone 
smoothly  in  the  reacting  room. 

A  second  experiment  exactly  repeats  the  first,  except  that  in 
it  the  commutator  rockers  are  thrown  over,  away  from  E,  and 
the  direction  of  current  in  the  chronoscope  magnets  is  thus  re- 
versed. With  a  little  practice,  the  experiments  run  their  course 
almost  automatically. 

The  direction  of  the  commutator  rockers  in  the  first  experiment  of  a  series 
should  always  be  noted  on  the  record  sheet.  In  the  present  instance,  the 
word  '  in '  would  be  written  below  the  column  of  figures,  since  we  have 
assumed  that  in  the  first  experiment  the  rockers  are  pulled  down  towards  E. 
Had  they  been  thrown  over,  the  word  '  out '  would  have  been  written.  This 
rule  is  necessary,  since  E  may,  by  some  chance,  forget  the  manipulation  in 
a  given  experiment.  If  he  knows  that  all  the  odd-numbered  experiments 
are  <  ins,'  and  all  the  even-numbered  '  outs,'  or  conversely,  the  lapse  of 
memory  does  not  matter. 

We  may  now  look  at  the  experiment  from  (9's  point  of  view. 
After  seating  himself  at  the  reaction  table,  O  signals  twice  to 
E,  to  indicate  that  he  is  ready.  He  then  lays  his  right  arm  on 
the  cushion.  As  soon  as  he  hears  ^'s  signal,  he  closes  the 
reaction  key,  and  listens  for  the  sound  of  the  hammer.  Having 
reacted,  he  ticks  off  the  experiment,  by  its  number,  on  his  record 
sheet ;  or,  if  anything  has  gone  wrong  with  the  apparatus,  if 
his  attention  has  wandered,  etc.,  makes  a  brief  note  to  that  effect. 
He  then  signals  once  to  E.  Finally,  he  replaces  his  arm  on  the 
cushion,  and  passively  awaits  E^  second  signal. 

If  the  preliminary  work  has  been  properly  done  (and  it  is  ^'s  business 
to  do  it  properly),  the  apparatus  will  work  without  any  hitch,  and  (9's  signal 
will  simply  be  an  acknowledgment  to  E  that  an  experiment  has  been  made. 
It  is  very  important  that  E  and  O  keep  in  touch,  in  this  way,  from  experi- 


1/2  The  Reaction  Expeiiment 

ment  to  experiment ;  nothing  is  more  annoying  than  to  have,  say,  50  experi- 
ments recorded  in  the  one  room  and  only  49  in  the  other. 

No  more  elaborate  signalling  is  necessary,  and  none  should  be  attempted. 

Control  Experiments. — We  must  now  go  back  again  to  the 
apparatus.  Before  work  upon  reaction  times  is  begun,  the  clock 
must  be  regulated.     The  procedure  is  as  follows. 

The  contacts  of  the  control  instrument  are  carefully  set  to 
some  known  time:  say,  1500-.  The  reaction  key  is  closed,  by 
the  short-circuit  switch  ;  the  sound  hammer  is  also  closed,  either 
by  relaxing  the  spring  or  (more  conveniently)  by  slipping  a  block 
of  wood  between  the  hammer  head  and  the  restraining  arm  that 
projects  above  it.  The  galvanometer  is  switched  out  of  the 
clock  circuit ;  the  eccentrics  at  the  back  of  the  clock  are  turned 
down. 

E  pulls  down  the  rocker  of  i,  and  brings  up  the  control  ham- 
mer (or  screen  or  pendulum)  to  its  magnet.  He  then  pulls  down 
the  rocker  of  2,  and  starts  the  clockwork  by  a  pull  upon  the 
nearer  cord.  The  circuit  is  not  yet  closed,  and  the  pointers  of 
the  chronoscope  dials  do  not  move.  As  soon  as  the  clock  gives 
out  its  proper  tone,  E  opens  i,  and  the  hammer  falls.  The 
clock-hands  move  during  the  isocr  that  elapse  between  make 
and  break  of  the  control  contacts.  As  soon  as  they  stand  still, 
E  pulls  the  farther  cord,  and  stops  the  clockwork  ;  he  then  opens 
2.  The  time  recorded  by  the  chronoscope  is  noted.  The  ex- 
periment is  repeated,  with  reversal  of  the  rockers. 

The  results  of  these  two  trials  should  be  sensibly  the  same. 
If  the  time  recorded  is  more  than  1500-,  the  current  must  be 
weakened  (more  resistance  put  in  the  clock  circuit)  ;  if  it  is  less 
than  1500-,  the  current  must  be  strengthened  (resistance  taken 
out  of  the  circuit,  or  an  extra  cell  added  to  the  battery).  The 
current  is  to  be  regulated,  until  the  clock  records  an  average  1 5  co- 
in ten  successive  trials,  with  an  MV  oi  not  more  than  i.5<t. 
The  position  of  the  eccentrics  should  not  be  changed,  if  change 
can  be  avoided. 

When  the  right  strength  of  current  has  been  found,  the  con- 
trol instrument  is  closed,  and  the  galvanometer  switched  into  the 
clock  circuit.  The  reading  of  the  needle,  in  both  directions 
(both  positions  of  the   rocker  of    2),  is   carefully  noted.     This 


§  24-    The   Three   Types  of  Simple  Reaction  173 

reading-  must  be  reestablished,  by  help  of  the  adjustable  resis- 
tance of  the  rheochord,  for  every  series  of  reaction  experiments. 
So  long  as  the  tension  of  the  magnet  springs  and  the  strength  of 
current  remain  the  same,  the  clock  will  record  an  average  1500" 
as  the  equivalent  of  the  control  time. 

The  function  of  the  control  instrument,  after  this  initial  test, 
is  to  keep  a  check  upon  the  variable  error  of  the  chronoscope. 
There  are  a  good  many  connections  in  the  circuit,  and  in  course 
of  time  some  contacts  will  become  defective  or  some  connections 
wear  loose.  Or  some  of  the  instruments  may  be  required  for 
other  experiments  in  the  laboratory,  and  may  have  been  care- 
lessly replaced.  Hence  it  is  possible  that  the  clock  times,  even 
with  the  right  strength  of  current  as  shown  by  the  galvanometer, 
will  presently  become  irregular.  If  this  is  the  case, — if  the 
clock  times  show  an  MV  oi  more  than  i.Scr, — the  circuit  must 
be  overhauled,  and  the  defects  made  good. 

Programme  of  Work. — We  are  now  able  to  make  out  the 
programme  of  an  afternoon's  work.  ( i )  At  the  beginning  of  the 
laboratory  period,  the  current  is  adjusted  to  its  right  strength  by 
means  of  galvanometer  and  rheochord.  (2)  Ten  control  times 
are  then  taken  ;  they  give  an  average  of  1500-,  with  a,n  MVoi 
i.5cr  or  less.  The  reaction  work  proper  now  begins.  (3)  In  the 
middle  of  the  period,  ten  more  control  times  are  taken  ;  the  re- 
sult should  be  as  before.  The  reaction  work  is  then  continued. 
(4)  At  the  end  of  the  period,  ten  final  control  times  are  taken  : 
again,  the  result  should  be  as  before.  An  increase  of  the  MV 
indicates  that  the  circuit  needs  inspection  :  an  increase  or  decrease 
in  the  average  time  recorded  means  that  the  current  strength 
has  varied.  The  results  of  the  control  experiments  must  be 
entered,  in  full,  in  the  note-book,  along  with  the  results  of  the 
reaction  experiments. 

The  control  experiments  look  more  formidable  in  print  than  they  are  in 
fact.  If  the  source  of  supply  is  constant,  the  first  adjustment  of  the  current 
takes  only  a  couple  of  minutes  ;  and  the  time  required  for  30  control  ex- 
periments is  quite  insignificant. 

Another  Arrangement  of  Apparatus. — Fig.  yy  shows  a 
satisfactory  disposition  of  the  apparatus  in  arrangement  I  of 
the  reaction  circuit. 


174 


The  Reaction  Experiment 


E  sits  on  a  swivel  chair  directly  facing  the  chronoscope.  The 
commutator  of  the  reaction  circuit  (marked  2  in  the  Fig.)  is 
placed  to  the  left  of  the  chronoscope.      Wires    run  from   the 


Fig.  77. 

nearer  poles  of  the  commutator  to  the  chronoscope,  rheochord, 
and  (by  way  of  the  switch)  to  the  galvanometer  :  this  is  the 
high-resistance  circuit  A  of  the  arrangement.  From  the  farther 
poles,  wires  run,  by  way  of  the  control  instrument,  to  the  sound 
hammer  and  reaction  key  :  this  is  the  low-resistance  circuit  B. 
Rheochord  and  galvanometer  are  placed  to  the  right  of  the 
chronoscope,  within  easy  reach  of  E ;  the  control  instrument 
stands  on  a  separate  table  to  his  left ;  the  sound  hammer  and  re- 
action key  are  on  a  table  in  an  adjoining  room.  Commutator  3, 
which  controls  the  magnet  of  the  sound  hammer,  is  placed 
to  the  right  of  the  chronoscope;  commutator  i,  which  controls 
the  release  magnet  of  the  control  instrument,  is  placed  to  the 


§  24.    The   Three   Types  of  Simple  Reaction  175 

left  of  commutator  2.     The    remaining  details  of  the  arrange- 
ment will  be  understood  from  the  Fig. 

The  preliminaries  for  an  experiment  are  as  given  above,  p.  1 70. 
The  procedure,  on  ^'s  part,  is  as  follows,  (i)  £"  places  his  left 
hand  upon  the  rocker  of  2,  and  pulls  it  over  towards  him.  The 
chronoscope  circuit  is  now  closed  ;  the  hands  of  the  clock  will  not 
move,  even  if  the  clockwork  is  started.  (2)  With  the  same  hand, 
E  pulls  the  nearer  cord  of  the  chronoscope,  and  starts  the 
clockwork.  (3)  Reaching  out  with  the  right  hand,  he  presses 
the  button  of  the  signal  circuit.  (4)  After  a  short  2  sec,  he 
pulls  down  (with  the  same  hand)  the  rocker  of  3,  and  thus  gives 
the  reaction  stimulus.  The  clock-hands  immediately  begin  to 
move,  and  stop  only  on  the  breaking  of  contact  at  the  reaction  key. 
(5)  As  soon  as  they  stop,  E  pulls  the  farther  cord  of  the  chrono- 
scope (left  hand),  and  (6)  throws  open  2  and  3  (left  and  right 
hands).  Everything  is  now  as  it  was  before  the  experiment  was 
taken.  (7)  He  then  records  the  time,  as  shown  by  the  dials.  A 
signal  from  O  assures  him  that  affairs  have  gone  smoothly  in  the 
reacting  room. — A  second  experiment  repeats  the  first,  except  that 
in  it  the  rockers  are  thrown  over,  away  from  E.  For  6^'s  duties, 
see  above,  p.  171. 

Adjustment  of  Current  Strength. — Suppose  that  we  take  the  resistance 
of  A  to  be  200,  and  that  of  B  to  be  20  ohms.  We  know  that  a  current  of 
60  milliamperes  is  sufficient  for  the  clock  circuit  in  arrangement  II.  Let 
us  assume  that  the  same  current  suffices  for  the  present  arrangement.  We 
shall  then  require,  for  circuit  A^  a  pressure  of  12  volts. 

It  is  necessary,  now,  that  the  current  remain  constant  both  (i)  when  A 
alone  is  closed,  and  (2)  when  A  and  B  are  closed  together.  The  joint  re- 
sistance of  the  two  conductors,  whose  separate  resistances  are  200  and  20 
ohms,  is  approximately  18.2  ohms.  To  force  a  current  of  60  milliamperes 
against  a  resistance  of  18.2  ohms  requires  a  pressure  of  1.09  volts.  And 
of  the  60  milliamperes,  ^\°^  or  5.5  will  be  in  the  clock  circuit  A^  and  \\% 
or  54.5  in  the  low  resistance  circuit  B.  In  other  words,  under  the  conditions 
that  we  have  laid  down,  the  closure  of  both  A  and  B  will  mean  that  we  have 
in  the  clock  circuit  a  current  of  1.09  volts,  5.5  milHamperes. 

The  following  questions  now  arise.  Is  our  initial  assumption  (that  we 
can  work  with  a  current  of  12  volts,  60  milliamperes)  correct?  How  must 
we  arrange  our  battery,  to  secure  a  constancy  of  the  current  delivered  ( i ) 
for  A  alone,  and  (2)  for  A  and  B  together?  Is  a  current  of  1.09  volts, 
5.5  miUiamperes  so  small  as  to  leave  the  clock  unaffected  ?     Is  the  ratio  of 


1/6  The  Reaction  Experiment 

resistance  200  :  20  a  good  ratio  to  choose,  or  can  it  be  improved  upon  ? — 
These  questions  should  be  answered  both  theoretically  and  experimentally. 

Control  Experiments. — The  procedure  is  as  above,  p.  172,  except  that 
the  eccentrics  at  the  back  of  the  clock  are  turned  up.  The  clock-hands 
move  during  the  1500-  that  elapse  between  make  and  break  of  the  control 
contacts,  /,  e.,  between  make  and  break  of  circuit  B. 

The  clock  time  may  be  standardised,  mutatis  7nutandis,  as  in  arrange- 
ment II.  It  may,  however,  be  simpler  first  to  adjust  the  current  so  that  the 
apparatus  works  well,  and  then  to  use  the  control  instrument  as  a  control 
of  both  constant  and  variable  errors.  Suppose,  e.  g.^  that  with  a  given 
current  the  clock  records  160a,  with  an  MV  oi  1.50-  or  less.  We  may 
either  adjust  the  current  until  the  clock  records  1 500-,  or  we  may  leave 
things  as  they  are,  and  correct  our  reaction  times  after  the  event  for  a  con- 
stant error  of  +  locr.  In  the  latter  case,  we  use  the  galvanometer  merely  to 
indicate  the  strength  of  current  employed. — Consult  with  the  Instructor  as 
to  which  of  these  alternatives  is  to  be  adopted. 

Practice,  Fatigue,  etc. — The  effect  of  increasing  practice 
is  to  shorten  the  time  of  reaction,  and  to  reduce  its  variability. 
Both  E  and  O  have  had  so  much  practice  in  reaction  work  as  is 
required  by  Exp.  XXVI.,  vol.  i.,  117  ff.  Under  these  circum- 
stances, w,e  may  assume  that  a  fairly  constant  level  of  practice 
will  be  attained  if  one  whole  afternoon  is  devoted  to  practice 
experiments.  These  experiments  must  be  made  as  carefully 
and  conscientiously  as  the  experiments  that  are  to  be  used  for 
computation  ;  their  results  are  to  be  entered  in  the  note-book. 

The  effect  of  increasing  fatigue  is  to  lengthen  the  time  of 
reaction,  and  also  to  increase  its  variability.  To  avoid  this  source 
of  error,  we  make  the  experimental  series  short,  and  interpose 
periods  of  rest  between  series  and  series. 

However  practised  O  may  be,  he  comes  a  little  strangely  or 
unpreparedly  to  each  new  series.  One  or  two  experiments  must 
be  taken  before  he  *  warms  up '  to  the  work,  or  ^  gets  into  the 
swing '  of  the  procedure.  This  fact,  together  with  what  we 
have  said  of  the  influence  of  fatigue,  makes  it  advisable  to  begin 
the  work  with  series  of  1 3  experiments ;  to  leave  5  min.  pauses 
between  series  and  series  ;  and  to  discard  the  first  3  experi- 
ments of  each  series  for  purposes  of  computation. 

The  3  discarded  experiments  are,  of  course,  to  be  entered  in  the  note- 
book.    If  the  work  is  long  continued,  so  that  0  attains  a  fairly  high  level 


§  25.    The   Three   Types  of  Simple  Reaction  177 

of  practice,  the  series  may  be  increased  to  22,  of  which  only  the  first  2  ex- 
periments are  discarded. 

Treatment  of  Results. — ( i )  The  apparatus  which  we  em- 
ploy in  the  reaction  experiment  is  to  work  within  certain  limits 
of  error  (i  i-Str,  in  our  illustrations),  and  is  to  be  manipulated 
in  a  certain  time-order  and  with  a  certain  time-interval  between 
signal  and  reaction  stimulus.  If  these  rules  are  infringed,  the 
result  of  the  experiment  is  worthless,  and  must  be  discarded. 

It  may  happen,  e.  g.^  that  E  is  disturbed,  and  forgets  to  give  the  signal ; 
or  that,  having  given  the  signal,  and  noticing  that  the  clock  has  nearly  run 
down,  he  curtails  the  2  sec.  interval  in  his  anxiety  to  save  the  experiment ; 
or  that  a  change  in  the  tone  of  the  regulating  spring  occurs  after  the  signal 
has  been  given ;  etc.,  etc.  Under  such  circumstances,  the  faulty  experiment 
is  noted,  but  its  result  is  discarded,  and  the  series  is  lengthened  by  one  ex- 
periment. The  case  becomes  more  serious  if  two  or  three  disturbances  of 
the  kind  appear  in  the  course  of  a  1 3-experiment  series.  At  best,  the  inser- 
tion of  the  additional  experiments  required  to  give  10  valid  results  is  likely 
to  fatigue  Oj  and  if,  further,  O  himself  is  led  to  suspect  that  E  is  careless, 
the  reactions  will  not  be  made  with  the  necessary  sustained  attention,  £"'3 
wisest  plan  is  to  cancel  the  series  ;  to  explain  to  O  that  something  has  gone 
wrong  with  the  instruments ;  and  to  inspect  the  circuits  and  practice  the 
manipulations  over  again.  It  need  hardly  be  repeated  that,  with  decent 
care,  the  experiments  will  run  smoothly  and  mistakes  occur  but  rarely. 

Shall  we  discard,  in  the  same  way,  the  results  of  experiments 
which  O  declares  by  introspection  to  be  worthless }  No !  We 
must  keep  a  record  of  (9's  introspections,  when  he  gives  them  ; 
but  we  must  not  throw  out  any  of  his  results.  The  object  of  the 
experiment  is  to  determine,  in  quantitative  terms,  (9's  mode  of 
behaviour  under  certain  fixed  conditions.  O  is,  for  the  time  being, 
a  psychophysical  machine,  a  reacting  machine  ;  and  it  is  a  part  of 
our  purpose  to  discover  the  variations  to  which  such  a  human 
machine  is  liable.  We  are  not  now  attempting  the  analysis  of 
the  action  consciousness ;  the  introspections,  valuable  as  they 
may  be,  are  only  incidental :  we  are  in  search  of  6^'s  reaction 
time^  and  of  the  manner  and  limits  of  its  variability.  The  fact 
that  O  occasionally  makes  mistakes,  disobeys  instructions,  gives 
extremely  high  or  extremely  low  time  values,  is  a  feature  of  his 
reacting  behaviour  which  we  must  record,  and  allow  its  due  place 

L 


178  The  Reaction  Experiment 

in  our  final  computation.     We  must  not  overemphasise  it ;  but 
neither  must  we  ignore  it. 

(2j  A  complete  statement  of  the  results  of  the  reaction  experi- 
ment includes  a  Hst  of  the  different  values  obtained,  and  a  record 
of  the  number  of  times  that  each  value  was  found.  Such  a  list 
and  record  are  called,  together,  a  table  of  frequencies  or  a  dis- 
tribution. The  following  Table,  e.  g.y  shows  the  distribution  of 
400  reaction  times. 


It  is  important  here  to  recall  the  fact  that  in  measuring  a  quantity  like 
reaction  time  we  are  measuring  to  the  nearest  scale  mark, — with  the  Hipp 
chronoscope,  to  the  nearest  y^Vir  sec.  Thus,  the  value  1580-  means  a 
value  that  is  nearer  to  1580-  than  it  is  either  to  157  or  to  159:  means,  /.  e., 
a  time  which  lies  between  157.5  and  158.50-. 

In  constructing  a  table  of  frequencies,  it  is  usual  to  take  the  unitscale- 
distances  somewhat  wider.  In  the  following  Table,  the  unit  is  5<r  :  so  that 
the  value  1 20  includes  all  recorded  times  between  1 20  and  1 24 ;  the  value 
125  all  times  between  125  and  129;  the  value  205,  all  times  between  205 
and  209. 


TABLE    OF    DISTRIBUTION    OF    REACTION   TIMES. 

From  E.  L.  Thorndike,  Theory  of  Mental  and  Social  Measure- 
ments, 1904,  25. 


FlME  IN  (T 

Frequency 

Time  in  ct 

Frequency 

120 

9 

165 

18 

125 

18 

170 

24 

130 

35 

175 

II 

135 

37 

180 

15 

140 

43 

185 

20 

145 

36 

190 

10 

150 

38 

19s 

3 

155 

40 

200 

4 

160 

38 

205 

I 

Such  a  Table  may,  evidently,  be  represented  in  graphic  form. 


§25-    The   Three  Types  of  Simple  Reaction 


179 


The  times  are  marked  off,  along  a  horizontal  line,  as  abscissae  ;  the 
frequencies  are  denoted  by  columns  of  varying  height  erected 
upon  these  abscissae.  The  figure  thus  obtained  is  termed  a  sur- 
face of  frequency  or  a  frequency  polygon.  The  contour  of  the 
surface — the  compound  Hne  that  connects  the  tops  of  the  col- 
umns, together  with  the  horizontal  base  line — is  termed  a  cufve 
of  distribution.  Figg.  78-8 1  show  various  modes  of  constructing 
the  frequency  polygon  from  reaction  data. 


n 


IT)    in   m 

Ov     o    — 

—       tN      CM 


Fig.  78. 


Frequency  polygon  of  *  reaction  times '  of  frog  to  electrical  stimulation  (method  of 

rectangles).     Unit  of  abscissa5=:i0(r.     Number  of  experiments=ioo.     From 

R.  M.  Yerkes,  Psych.  Bulletin,  i.,  1904,  140. 
Note  that  the  abscissal  numbers  stand  in  the  middle  of  their  unit  distances.     This 

means  that  the  class  represented  by  a  number  extends  both  below  and  above  it. 

Thus,  135  is  the  class  number  containing  all  times  between  130  and  139. 


i8o 


The  Reaction  Experime7tt 


-L-J I I \ LU I I      I      I      I      ) 


L20  123  130  135  14.0  14.5  150  155  160  165  170  175  180  185  190  195  200  205  210 


Fig. 


79- 


Frequency  polygon  of  the  reaction  times  listed  in  the  Table  on  p.  178  (method 
of  rectangles).     Unit  of  abscissae=5<r.     Number  of  experiments=400. 

Note  that  the  abscissal  numbers  stand  at  the  beginning  of  the  unit  distances. 
This  means  that  the  class  represented  by  a  number  extends  only  above  it. 
Thus,  120  is  the  class  containing  all  times  between  120  and  124. 


120  125  130  135  140  145  150  155  160  165  170.175  J80  185  190  195  200  205  210 


Fig.  80. 
Frequency  polygon  of  the  reaction  times  listed  in  the  Table  on  p.  178  (method  of 
trapezia).     Unit  of  abscissae  and    number  of  experiments  as  in  Fig.  79.      The 
figure  is  obtained  by  joining  the  tops  of   the  middle  ordinates  of  successive 
contiguous  rectangles. 


§25-    The   Three   Types  of  Simple  Reaction  i8i 


6^7     89  100  1/234     56     7     89  200  1     234567     89  300  1     2     3 

Fig.  8i. 

Frequency  polygons  of  A  muscular,  B  central,  and  C  sensorial  reactions  to  light. 
Unit  of  abscissae:=iO(r.  Number  of  experiments  in  each  case:=i5o.  From 
N.  Alechsieff,  Philos.  Studien,  xvi.,  1900,  Taf.  I. 

The  abscissal  numbers  stand  at  the  beginning  of  the  unit  distances  ;  so  that  the 
first  class,  e.  g.,  comprises  all  times  between  70  and  790-.  Above  the  abscissal 
numbers  ordinates  have  been  erected,  whose  successive  heights  are  proportional 
to  the  frequency  of  the  classes  (method  of  loaded  ordinates).  This  method 
of  construction  applies  in  strictness  only  to  integral,  not  to  graduated  variates : 
hence  the  method  of  Fig.  80  should  have  been  employed. 

(3)  It  has  been  customary,  in  psychophysics,  to  use  the  arith- 
metical mean  or  average,  and  the  mean  variation  or  average 
deviation,  as  the  representative  measures  of  the  reaction 
time  and  its  variability.  The  mean  of  any  set  of  data  rep- 
resented by  a  frequency  polygon    of    the    type  of   Fig.    '/^   is 

I(r.f) 
determined  by  the  formula  M  =  — — — -,  in  which  r  is  the  reac- 

n 

tion-time  of  any  class,  /  its  frequency,  I  indicates  that  the  sum  of 

the  products  for  all  classes  into  frequency  is  to  be  obtained,  and  n 

is  the  number  of  reactions  taken.     Thus  the  mean  of  the  frog 

reactions  represented  in  Fig.  /Sis  (135x6+145x6+155x6  + 

165  X  11  +  175x15  +  185x25  +  195x12+205x12+215x6+ 

2  3  5  X  I ) -M  00=  I  So.oocr. 

If  the  frequency  polygon  have  the  form  of  Fig.  79,  in  which 


iSz  The  Reaction  Experiment 

each  abscissal  number  means  '  the  numbers  from  this  figure  to 

the  next  on  the  scale,'  then  the  mean  calculated  from  it  must,  if 

it  is  to  represent  an  absolute  point  on  the  scale,  be  increased  by 

half  the  unit  of  the  scale. 

The  mean  variation  is  determined  from  the  formula  MV^=^ 

lid.f) 

— ,  where  d=  deviations  of  class  from  mean,  and  /,  i",  n  have 

n 

the  meanings  assigned  to  them  in  the  previous  formula.     Thus 

the  mean  variation  of  the  frog  reactions  is   (45x6+35x6  + 

25x6+15x11  +  5X15  +  5X25  +  15X12  +  25x12  +  35x6  +  55 
X  i)+- 100=1 7.400-. 

(4)  It  is  important  to  state,  not  only  the  mean  and  the  mean 
variation,  but   also  the   relative  variability  of  an   experimental 

series.     This  is  obtained  by  the  formula  rv  = .^^°°.      Its    im- 

portance  lies  in  the  fact  that  the  J/ F  depends  not  only  on  the 
deviations  of  the  individual  reaction  times  from  the  mean,  but  also 
on  the  actual  magnitude  of  the  mean  itself.  Thus  an  O  who  re- 
acts to  sound  in  I20cr,  with  a.nMVoi  lOcr,  and  to  light  in  i8ocr, 
with  an  MV  oi  150-,  is  reacting  with  the  same  degree  of  con- 
stancy in  each  case,  although  the  absolute  values  of  the  J/ Fare 
different. 

The  rv  of  the  frog  experiments  is  -^W"*  ^^  9.66cr. 

(5)  Lastly,  it  is  important  to  state  the  range  of  the  results, 
i.  e.,  the  extreme  times  between  which  the  series  of  reactions 
varies.      In  the  case  before  us,  these  times  were  133  and  232cr. 

The  note-book  record  now  includes  the  following  data :  {a) 
the  crude  results,  with  (?'s  occasional  introspections ;  {b)  a  table 
of  frequencies,  at  the  head  and  foot  of  which  stand  the  extreme 
values  representing  (c)  the  range  of  the  series ;  {d)  a  frequency 
polygon  ;  and  {e)  the  average  of  the  series,  with  (/)  its  mean  va- 
riation and  [g)  its  relative  variability. 

A  full  statistical  treatment  of  the  results  implies  the  determi- 
nation of  other  representative  values.  Thus,  besides  the  My  it 
may  be  worth  while  to  determine  {k)  the  median,  and  possibly 
(/')  the  mode  of  the  series.  Besides  the  MV,  we  may  determine 
(/)  the   standard  deviation  of  the  M.     Besides  the  rv,  we  may 


§  25.    The   Three   Types  of  Simple  Reaction  183 

determine  {k)  the  coefficient  of  variation  of  the  series.  And  we 
may  also  determine  the  probable  error  (/)  of  the  mean  and 
{in)  of  the  standard  deviation.  There  is  a  long  list  of  such  values  ; 
but  these  are  the  most  important  for  psychophysical  purposes. 
The  Instructor,  if  he  thinks  that  you  should  determine  them, 
will  furnish  you  with  definitions  of  the  terms,  with  the  required 
formulae,  and  with  references  for  reading.  Be  sure  that  you 
thoroughly  understand  the  significance  of  the  values  before  you 
enter  upon  their  computation.  It  is  a  simple  waste  of  time  to 
work  out  a  column  of  numerical  results  which  you  do  not  com- 
prehend, and  which  you  could  not  use  if  opportunity  for  their 
use  occurred. 

Further  Experiments. — The  Instructor  will  determine  the 
number  of  sound  reactions  to  be  taken.  When  these  have  been 
completed,  new  series  of  experiments  are  to  be  begun  with  light 
and  pressure  stimuli. 

[a)  A  light  pendulum,  e.  g.^  may  replace  the  sound  hammer  on  (9's  table  : 
it  should  stand  sidewise  to  a  window,  and  the  light  should  be  reflected 
through  the  slit  in  the  screen  from  a  plane  mirror.  If  there  is  but  one  elec- 
tromagnetic release,  at  the  one  end  of  the  pendulum  arc,  so  that  the  pendu- 
lum swings  back  and  forth  in  a  single  movement  and  is  caught  on  the  re- 
turn, the  rocker  of  commutator  3  should  be  thrown  completely  over  in 
each  experiment :  the  pendulum  leaves  the  magnet  at  break,  while  the 
remaking  of  the  circuit  actuates  the  magnet  in  time  for  the  return  of  the 
bob.  If  there  are  two  electromagnets,  so  that  the  pendulum  is  caught  at 
the  conclusion  of  a  half-swing,  and  has  to  be  brought  back  again  to  its 
starting-point  before  a  second  experiment  can  be  made,  the  commutator  is 
opened  (for  release  of  the  pendulum),  closed  again  on  the  same  side 
(the  catch  at  the  end  of  the  half  swing),  and  the  rocker  is  then  thrown  over 
(for  release  and  return  to  starting-point).  With  a  little  trouble  in  the  ad- 
justment of  current  strength,  papering  of  the  armature  surfaces,  etc.,  the 
pendulum  can  be  made  to  work  accurately,  and  practically  without  noise. 

{b)  The  use  of  the  ordinary  pressure  stimulators  implies  that  a  second  E 
sit  with  O  in  the  reacting  room.  If  electric  stimulation  or  the  stimulator 
called  for  by  Question  6,  p.  167  be  employed,  this  complication  is  avoided. 

The  Instructor  will  decide  whether  or  not  smell  and  taste  reac- 
tions are  to  be  taken  at  this  stage. 

Questions. — E  and  O  {i)  What  is  it  that  we  are -measuring  or 
determining,  in  the  reaction  experiment }  \  :i  r:  t  :;i.r .    : 


184  -^^^^^  Reaction  Experiment 

(2)  The  reaction  experiment  has  more  than  one  claim  to  be  in- 
cluded in  a  Course  in  Experimental  Psychology.  What  are  the 
various  reasons  for  its  introduction  ? 

(3)  Criticise,  in  connection,  the  following  statements  : 

{a)  "The  variations  in  the  length  of  the  reaction  time  are  most  strik- 
ing in  the  curves  of  the  sensorial  reactions.  That  is  to  say,  the 
greater  the  number  of  mental  processes  introduced  into  the  course 
of  the  reaction  experiment,  the  greater  is  the  variation  which  the 
curve  shows.  Evidently,  then,  more  weight  is  to  be  laid  upon 
these  variations  than  upon  the  absolute  time.  For,  from  the  stand- 
point of  psychological  analysis,  variations  that  are  thus  conditioned 
upon  psychological  factors  are  more  important  than  is  the  establish- 
ment of  average  values"  (Alechsieff,  1900). 

(J))  "  Variations  in  the  length  of  reaction  time  are  usually  dealt  with 
merely  in  their  bearing  on  the  trustworthiness  of  the  average  value. 
But  if  we  were  better  acquainted  experimentally  with  the  many  ele- 
ments which  enter  as  determining  factors  into  these  variations,  we 
might  be  able  to  make  a  much  more  extensive  use  than  we  can  do 
at  present  of  the  reaction  process  as  an  index  of  the  activities  of  the 
central  nervous  system"  (Smith,  1903). 

{c)  "  Variability,  or  the  degree  of  constancy  with  which  a  reaction  oc- 
curs, is,  for  certain  purposes,  of  equal  value  with  the  average  reac- 
tion time.  .  .  .  Any  or  all  of  these  values  [the  statistical  deter- 
minations mentioned  above,  pp.  1 82  f .]  might  be  useful  in  a  study  of 
the  series  of  [frog]  reaction  times  under  consideration"  (Yerkes, 
1904). 

(4)  "  Es  ist  nicht  entschieden  genug  zu  betonen,  dass  diese 
Versuche,  wie  sie  nach  ihrer  physiologischen  Bedeutung  die  com- 
plicirtesten  sind,  so  auch  an  die  psychologische  Befahigung  und 
Uebung  des  O  die  grossten  Anforderungen  stellen.  Sporadisch 
Oder  an  beliebig  ungeiibten  Personen  angestellte  Versuche  oder 
aber  endlich  solche,  bei  denen  man  ohne  sorgfaltige  Controle 
durch  die  Selbstbeobachtung  planlos  Reactionszeiten  misst,  sind 
daher  werthlos  "  (Wundt,  1903).  This  criticism  would  rule  out  of 
Experimental  Psychology  the  investigations  in  which  the  reaction 
experiment  has  been  employed  as  a  *  mental  test ' ;  a  good  deal 
of  work  done  upon  (9's  who  are  mentally  deranged ;  certain  an- 
thropological work  (experiments  upon  Indians,  Negroes,  etc.); 
and  all  experiments  performed  with  the  lower  animals.  Do  you 
subscribe  to  it }    Why  ? 


§  26.    Compound  Reactions ;  Discrimination  185 

(5)  Suggest  other  experiments  upon  action  that  may  lead  to 
psychological  results,  or  to  the  establishment  of  psychophysical 
constants. 

§  26.  Compound  Reactions ;  Discrimination,  Cognition  and  Choice. 

— If  the  "  reaction  consciousness  is  the  laboratory  form  of  the 
action  consciousness  of  everyday  life "  (vol.  i.,  118),  it  should 
be  possible  for  us  to  pass  beyond  impulsive  action  (the  simple 
reaction),  and  to  reproduce  under  experimental  conditions  the 
more  complex  actions  that  derive  from  the  impulse.  This  step 
has,  in  fact,  been  taken.  '  Compound  reactions,'  as  they  are 
termed,  were  first  made  by  the  Dutch  oculist  F.  C.  Bonders 
(18 18- 1889)  and  his  pupils,  and  have  since  been  studied  under 
very  various  forms.  The  principal  modes  of  procedure,  looked 
at  from  the  outside,  from  the  point  of  view  simply  of  stimulus 
and  movement,  may  be  tabulated  as  follows. 

I.    Experiments  with  one  stimulus  and  one  movement,  both  known  to  O : 
the  simple  reaction  (Bonders'  ^-method). 

II.     Experiments  with  more  than  one  stimulus  and  one  movement : 

{a)  the  stimuli  are  known   beforehand     to    O   (sub-form   of   Wundt's 

(/-method) ; 
{b)  the  stimuli   are  known  beforehand  to   O  only  in  the  most  general 
way  (type  of  Wundt's  ^-method). 

III.  Experiments    with    more    than    one    stimulus    and    more  than  one 

movement : 
{a)  the  stimuli  and  the  correlated  movements  are  known  beforehand  to 

O  (sub-form  of  Bonders'  ^-method)  ; 
{b)  the  stimuli  and  the  movements  are  known  beforehand  to  O  only  ia 

the  most  general  way  (type  of  Bonders'  /^-method). 

IV.  Experiments  with  more  than  one  stimulus  and  with  the  alternative  of 

movement  and  no-movement : 
{a)  the   stimuli    are   known  beforehand  to  O    (sub-form  of  Bonders* 

^-method)  ; 
{b)  the  stimuli    are  known  beforehand  to  O  only  in  the  most  general 

way  (type  of  Bonders'  ^-method). 

These  four  methods  may  be  illustrated  as  follows  : 

Bonders'   ^-method :     the  stimulus  is  the  spoken  syllable  koj  O  responds 
by  uttering  the  same  syllable. 


1 86  The  Reaction  Experiment 

Bonders'  <5-method  :  the  stimulus  is  a  spoken  syllable  made  up  of  k 
and  a  following  vowel ;  O  responds  by  uttering 
the  stimulus-syllable. 

Bonders'  <;-method  :  the  stimulus  is  a  spoken  syllable  made  up  of  k 
and  a  following  vowel ;  O  responds  only  to  the 
stimulus  ko  by  uttering  ko. 

Wundt's  ^-method  :  the  stimulus  is  a  spoken  syllable  made  up  of  k  and 
a  following  vowel ;  O  responds  to  each  stimulus, 
as  he  apprehends  it,  by  uttering  ko. 


We  need  say  nothing  more  about  the  <^-method,  which  has 
been  discussed  in  the  previous  §§.  Nor  shall  we  say  anything 
here  of  another  type  of  compound  reaction,  the  7'eaction  on  as- 
sociationy  in  which  the  reaction  method  is  turned  to  account  for 
the  study  of  the  associative  consciousness  (see  §27  below). 
We  must,  however,  look  a  little  more  closely  at  methods  II., 
III.,  and  IV.  from  the  inside,  from  the  point  of  view  of  the  men- 
tal processes  involved. 

II.  {a)  The  Discriminative  Reaction. — Suppose  that  you  are 
giving  simple  reactions  to  light.  You  see  a  flash  of  white  light, 
as  the  pendulum  swings,  or  a  bright  white  disc,  as  the  shutter 
opens.  Every  experiment  brings  back  the  same  stimulus ;  you 
know  always  what  to  expect.  Suppose,  on  the  contrary,  that 
you  are  told  to  expect  eitJier  white  or  black.  The  pendulum 
screen  or  the  shutter  is  faced  with  grey,  and  E  may,  at  his 
pleasure,  place  a  black  or  a  white  card  behind  the  opening. 
Your  reaction  has  now  become  a  more  complicated  matter.  You 
take  up  a  different  mental  attitude  to  the  whole  experiment : 
before,  you  were  definitely  expecting  a  certain  stimulus;  now, 
you  are  expecting  one  or  other  of  two  possible  stimuli.  You  take 
up  a  different  attitude  to  the  stimulus  itself,  as  it  is  presented : 
before,  you  took  its  quality  for  granted  ;  now  you  have  to  dis- 
tinguish it,  as  '  the  black  '  or  *  the  white  '  stimulus.  You  do  not 
name  it,  of  course  :  that  would  introduce  a  verbal  association  into 
the  experiment :  but  you  receive  it  discriminatively.  Lastly,  you 
take  up  a  different  attitude  to  the  reaction  movement  :  before, 
the  movement  followed  naturally  and  inevitably,  touched  off  by 
the  stimulus ;  now,  there  is  a  restraint  upon  the  movement,  you 
dare  not  lift  your  finger  till  you  have  satisfied  yourself  of  the 


§  26.    Compound  Reactiojts ;  Discrimination  187 

quality  of  the  stimulus.  Reactions  of  this  sort  are  called  dis- 
criminative reactions,  and  the  times  are  called  discrimination 
times  (temps  de  distinction,  Unterscheidungszeiten) . 

Suggestions  for  Work  upon  Discriminative  Reactions. — 
(i)  Two  visual  stimuli :  black  and  white,  or  red  and  green,  or  blue 
and  yellow.  (2)  Four  visual  stimuli :  black  and  white,  and  two 
colours.  (3)  Six  visual  stimuli :  black  and  white,  and  four  colours. 
All  these  experiments  can  be  performed  with  the  visual  stimula- 
tors described  above  :  pp.  156  f.  (4)  Two  or  more  light  inten- 
sities :  greys  may  be  used  in  the  fall  chronometer  or  shutter ; 
various  thicknesses  of  tissue  paper  in  the  pendulum.  (5)  Noise 
and  clang  :  place  a  single-stroke  bell  along  with  the  sound  ham- 
mer in  the  reaction  circuit,  so  that  either  instrument  may  be 
sv/itched  in  at  will.  (6)  Two  or  three  intensities  of  noise :  these 
may  be  obtained  by  regulating  the  spring  of  the  sound  hammer. 
(7)  Two  or  three  clangs  of  different  pitch  :  use  Martins'  arrange- 
ment of  monochord  and  pick.  (8)  Mechanical  pressure  and 
electrical  stimulation  :  two  alternative  circuits  are  arranged,  as 
for  noise  and  clang  ;  the  electrodes  are  attached  permanently 
to  (9's  hand,  and  a  stimulus  is  given  sometimes  by  them,  some- 
times by  the  sensibilometer.  (9)  Two  intensities  of  mechanical 
pressure  or  of  electrical  stimulation.  (10)  Sounds  to  right  and 
left  :  the  two  sound  hammers  are  set  to  give  stimuli  of  equal  in- 
tensity.    (11)    Pressures  or  electrical  stimuli  to  right  and  left. 

II.  {h)  The  Cognitive  Reaction.— h\  the  discriminative  re- 
action, the  stimuli  employed  are  always  known  beforehand  to 
O  ;  the  colours  are  shown,  the  sounds  presented,  before  the  reac- 
tions are  taken.  In  the  cognitive  reaction,  the  stimuli  are  known 
only  in  a  very  general  way.  O  is  told,  e.  g.y  that  he  will  be  shown 
a  simple  visual  impression,  a  colour  or  a  brightness ;  and  he  is  not 
to  react  until  he  has  *  cognised,'  identified,  apprehended  the  par- 
ticular quality  of  the  stimulus.  The  cognitive  reaction  is  thus, 
in  the  rough,  a  more  general  form  of  the  discriminative  reaction. 
The  preliminary  expectation  is  less  definite ;  the  stimulus  is 
received,  not  discriminatively,  but  as  a  famihar  object  presenting 
itself  for  identification ;  the  movement  is  restrained  until  iden- 
tification  is  complete.     The  results  of  the  experiment,  on  the 


i88 


TJie  Reaction  Experiment 


objective    side,  are  called  cognition  times   (temps  de  reconnais- 
sance, Erkennungszeiten). 

Suggestions  for  Work  upon  Cognitive  Reactions. — 
Visual  stimuli  offer  the  best  material  for  these  reactions.  We 
may  take  (i)  colours  and  brightnesses;  (2)  single  letters ;  (3) 


e 


^ffi 


-rit>     at. 


Fig. 
Reading  telescope. 

short  words.  The  type  used  for  the  letters  and  words  should  be 
that  known  as  Gothic,  grotesque  or  sansserif  (without  serifs  and 
hair-lines);  the  stimulus  is  observed,  not  with  the  naked  eye,  but 
through  a  reading  telescope.  We  may  also  employ  (4)  a  pro- 
gressive series  of,  say,  one  to  six  place  numbers. 


III.,  IV.  The  reactions  under  these  two  headings  are  gen- 
erally grouped  together  as  choice  reactions  (temps  de  choix, 
Wahlzeiten).     We  may  classify  them  in  detail  as  follows. 

III.  {a)  The  Selective  Reaction  with  Discrimination. — The 
reaction  is  discriminative  on  the  side  of  stimulus,  selective  on  the 
side  of  movement.  That  is  to  say,  the  various  stimuli  presented 
to  O  are  known  to  him  beforehand ;  he  receives  them  discrimi- 
natively:  while  he  is  also  called  upon  to  choose  between  the 
various  movements  which,  under  the  instructions  given  him,  are 
correlated  with  these  stimuli. 


§  26.    Compou7id  Reactions ;  Discrimination  189 

Suggestions  for  Work  upon  Selective  Reactions  (Dis- 
criminative Type). — The  simplest  form  of  discriminative  re- 
action is  that  in  which  two  known  stimuh  are  presented  to  O  in 
irregular  order.  Plainly,  then,  the  simplest  form  of  selective 
reaction  with  discrimination  will  be  that  in  which  the  two  known 
stimuli  are  replied  to,  the  one  by  a  right-hand,  the  other  by  a 
left-hand  movement.  Two  reaction  keys,  placed  conveniently  for 
O's  two  hands,  are  introduced  (in  series)  into  the  reaction  cir- 
cuit, and  the  reacting  movement  is  made  with  the  hand  to  which 
the  stimulus  presented  appeals.  All  the  modes  of  discriminative 
reaction  listed  above  may  thus  be  transformed  into  selective  re- 
actions. 

We  are  not,  however,  confined  to  a  choice  between  two  move- 
ments. We  may  use  a  five-finger  reaction  key,  and  react  to 
each  one  of  five  known  colours  by  the  movement  of  a  particular 
finger.     Or   we  may  take   two   five-finger  keys,  adapted  to  the 


Fig.  83. 
Five-finger  reaction  keys. 

right  and  left  hands  respectively,  and  react,  say,  to  the  stimuli  i, 
2,  3,  4,  5  by  movements  of  the  fingers  of  the  right  hand,  and  to 
the  stimuli  I,  II,  III,  IV,  V  by  movements  of  the  fingers  of  the 
left  hand. 

III.  [b)  The  Selective  Reaction  with  Cognition. — The  reaction 
is  cognitive  on  the  side  of  stimulus,  selective  on  the  side  of  move- 
ment. That  is  to  say,  the  stimuli  presented  to  O  are  not  known 
to  him,  beforehand,  except  in  a  general  way ;  he  receives  them 
cognitively  :  while  he  is  also  called  upon  to  choose  between  the 
various  movements  which,  by  a  natural  co5rdination,  are  corre- 
lated with  these  stimuli. 

Suggestions  for  Work  upon  Selective  Reactions  (Cog- 
nitive Type). — Since   the  stimuli  are  unknown  to  <9,  we  cannot 


IQO  The  Reaction  Experiment 

settle  beforehand  what  reaction  movements  are  to  be  connected 
with  them  ;  we  must  rather  look  about  us  for  a  set  of  movements 
which  are  naturally  coordinated  with  various  classes  of  sense  im- 
pressions. And  we  find  what  we  require  in  the  movements  of 
articulate  speech  ;  it  is  entirely  natural  to  us  to  name  the  objects 
of  our  surroundings.  We  accordingly  choose  as  stimuli  colours 
and  brightnesses,  or  letters,  or  short  words,  or  simple  drawings 
in  black-and-white ;  and  we  allow  O  to  react  by  speaking  the 
name  of  the  colour,  etc.,  into  a  voice  key. 

IV.  (a)  The  Volitional  Reaction  with  Discrimination. — The 
reaction  is  discriminative  on  the  side  of  stimulus,  volitional  on  the 
side  of  movement.  The  stimuli  are  known  to  O ;  he  receives 
them  discriminatively  :  while  the  reaction  movement  is  correlated 
only  with  certain  members  of  the  group,  the  rest  being  allowed 
to  pass  by  unregarded. 

Suggestions  for  Work  upon  Volitional  Reactions  (Dis- 
criminative Type.) — The  conditions  for  these  reactions,  in  their 
simplest  form,  are  identical  with  those  of  the  discriminative  re- 
action. We  select  two  stimuli, — say,  black  and  white ;  and  we 
instruct  O  to  react  to  the  black  by  lifting  his  finger  from  the 
key,  but  not  to  react  to  white  at  all.  All  the  modes  of  discrim- 
inative reaction  listed  above  may  thus  be  transformed  into  voli- 
tional reactions. 

We  are  not,  however,  confined  to  a  choice  between  a  single 
movement  and  no-movement.  We  may,  e.  g.^  take  two  five- 
finger  keys,  and  react  to  the  stimuH  i,  2,  3,  4,  5  by  movements  of 
the  fingers  of  the  right  hand,  and  to  the  stimuh  I,  II,  III,IV, 
V  by  movements  of  the  fingers  of  the  left  hand,  introducing  at 
irregular  intervals  a  stimulus  of  another  kind  (+  or  A)  to  which 
no  responsive  movement  is  to  be  made. 

IV.  {b)  The  Volitional  Reaction  with  Cognition, — The  reac- 
tion is  cognitive  on  the  side  of  stimulus,  volitional  on  the  side  of 
movement.  The  stimuli  are  known  to  O  only  in  a  general  way  ; 
he  receives  them  cognitively  :  while  the  reaction  movements  nat- 
urally co5rdinated  with  the  stimuli  are  in  some  cases  to  be  made, 
in  others  to  be  suppressed. 

Suggestions  for  Work  upon  Volitional  Reactions 
(Cognitive  Type). —  0  may  be  told,  e.  g.y  that  he  will  be  shown 


§  26.    Compound  Reactions ;  Discrimination  191 

either  a  colour  or  a  brightness ;  and  that  he  is  to  react  to  all  col- 
ours by  naming  them,  but  not  to  react  to  the  brightnesses  at  all. 
Or  he  may  be  told  that  he  will  be  shown  a  mixed  series  of  one- 
syllable  verbs  and  substantives ;  and  that  he  is  to  react  to  the 
verbs  by  uttering  them,  but  not  to  reply  to  the  substantives. 

EXPERIMENT  XXV 

The  Instructor  will  furnish  directions  for  this  experiment. 

Questions  and  Exercises. — E  and  O  {i)  Make  a  Table,  in 
the  form  of  a  genealogical  tree,  of  the  more  complex  types  of 
action  and  their  derivatives.  Give  an  account  of  the  composition 
of  the  motive  in  each  case. 

0  (2)  Give  a  full  analysis  of  the  cognition  and  the  discrimi- 
nation, as  mentally  experienced,  and  indicate  the  place  of  these 
complexes  in  the  psychological  system. 

E  (3)  Draw  a  diagram,  showing  the  disposition  of  apparatus 
in  some  one  of  the  following  experiments.  Does  the  experiment, 
as  you  arrange  it,  require  two  E^  or  one }  If  the  former,  can 
you  simplify  the  conditions,  or  suggest  other  apparatus,  so  that 
but  one  E  shall  be  necessary } 

(a)  Discriminative  reaction  no.  8  :  mechanical  pressure  and 

electrical  stimulation  ;  ordinary  reaction  key. 
{b)  Cognitive  reaction  no.  2  :  Hght  pendulum  (with  two  re- 
lease magnets)  and  reading  telescope  :  ordinary  reaction 
key. 
(c)  Selective  reaction  with  discrimination :  exposure  appara- 
tus for  ten  figures  ;  two  five-finger  reaction  keys. 
{k)  Selective  reaction  with   cognition  :  exposure  apparatus 
for  short  words ;  voice  key 
O  (4)  In  performing  experiments  upon  compound  reactions, 
you  are  called  upon  to  introspect.     The  action  consciousness  is, 
however,    much  more  complicated  than  it  was  in  the  simple  re- 
action experiment.     Can  you  suggest  any  plan  whereby  the  task 
of  introspection  shall  be  simplified  } 

O  (5)  In  the  discriminative  reaction  (p.  1 86)  you  are  not  al- 
lowed to  name  the  stimulus ;  in  the  selective  reaction  with  cog- 
nition you  react  by  naming.     Why  is  this  1 


192  The  Reaction  Experiment 

E  (6)  What  preliminary  work  must  be  done  before  selective 
reactions  with  discrimination  are  taken  ? 

O  (7)  Analyse  the  process  of  *  choice,'  with  reference  to 
selective  and  volitional  reactions. 

§  27.  Compound  Reactions:  Association. — In  the  reaction  ex- 
periments which  we  have  so  far  described,  O  reacts  after  he  has 
sensed,  discriminated  or  cognised  the  given  stimulus.  In  the 
associative  reaction,  he  does  not  react  to  the  given  stimulus  at 
all ;  he  makes  the  movement  of  reaction  only  after  the  stimulus 
has  suggested  something  else,  has  aroused  in  consciousness  some 
associated  idea.  It  is  this  associated  idea  to  which  the  move- 
ment is  the  response.  Suppose,  e.  g.,  that  we  are  working  with 
a  visual  exposure  apparatus  and  with  the  ordinary  finger  key. 
The  word  sea  is  shown.  Instead  of  reacting  as  soon  as  he  has 
cognised  the  stimulus,  O  waits  for  the  appearance  in  conscious- 
ness of  some  related  idea.  Let  this  be  the  verbal  idea  '  shore.' 
Then  O  lifts  his  finger  from  the  key  as  soon  as  the  idea  of '  shore  ' 
has  taken  shape. 

It  is  clear  that  the  associative  reaction  may  be  varied  in  a  great 
many  ways :  as  regards  stimulus,  as  regards  the  rules  of  associa- 
tion, and  as  regards  response.  We  will  look  first  at  the  associa- 
tions. 

I.  (i)  The  obvious  procedure  to  begin  with  is  to  leave  O  en- 
tirely free  in  the  matter  of  association.  We  show  him  a  word ; 
he  is  to  react  when  the  word  has  suggested  something,  no  matter 
what.  The  word  sea  may  arouse  the  idea  of  land  or  water  or 
ships  or  some  particular  sea  or  some  particular  incident  at  sea, — 
anything  it  likes.  Associations  of  this  sort  are  termed,  techni- 
cally, free  associations.  (2)  We  may,  however,  limit  the  range 
of  possible  associations.  We  tell  O  that  we  shall  show  him  pic- 
tures of  familiar  objects,  and  that  he  is  to  think  of  a  part  of  the 
presented  object,  or  of  an  attribute  of  it ;  that  we  shall  show  him 
adjectives,  and  that  he  is  to  think  of  corresponding  substantives  ; 
verbs,  and  he  is  to  think  of  corresponding  adverbs  ;  class-names, 
and  he  is  to  think  of  instances  that  fall  under  them ;  and  so  on. 
Associations  of  this  sort  are  termed  ambigitons  or  partially  con- 
strained associations.     We  ask  O  a  question,  to  which  different 


2/.    Compound  Reactions ;  Association 


193 


answers  are  possible;  but  the  number  of  answers  is  limited,  and 
the  association  idea  must  fall  within  this  limit.  (3)  Or  again, 
we  may  ask  O  a  question  that  admits  of  but  a  single  answer. 
We  may  show  him  the  names  of  countries,  to  which  he  is  to 
associate  the  capital  cities ;  words,  which  he  is  to  translate  into 
French  or  German ;  figures,  which  he  is  to  add  or  multiply. 
Associations  of  this  sort  are  termed  consti^aincd  or  univocal 
associations.  (4)  Lastly,  in  the  place  of  simple  successive  associ- 
ation, we  may  take  more  complex  mental  processes ;  we  may, 
by  our  stimulus,  ask  a  question  of  O  which  requires  for  its  an- 
swer a  judgment.  The  arrangement  of  such  an  experiment  is 
explained  under  II.  (3)  below. 

11.  We  turn  to  the  variation  of  the  stimulus.  It  is  usual  to  em- 
ploy ( I )  fairly  simple  visual  stimuli :  colours,  short  words,  figures, 
drawings  in  black-and-white,  etc.  These  are  exposed  in  the 
same  manner  and  with  the  same  precautions  as  in  the  preced- 


Fig.  84. 
Roemer's  exposure  apparatus. 

ing  experiments.  Sometimes  (2)  spoken  words  form  the  stimuli : 
E  may  press  a  finger  key  at  the  moment  of  utterance,  and  so 
close  the  reaction  circuit  as  the  stimulus  is  given,  or  he  may  use 
a  lip  key,  which  makes  contact  at  the  beginning  of  the  movement 
of  articulation.  (3)  When  a  judgment  of  comparison  replaces 
the  association,  the  stimulus  consists  of  a  spoken  sentence  :  e.  g.y 
"  Which  is  the  greater  poet :  Tennyson  or  Browning  }  "  and  E 
presses  a  finger  key  as  he  utters  the  final  word. 


M 


194  T^^^^  Reaction  Experiment 

III.  The  movement  of  reaction  may  be  (i)  the  lift  of  the 
finger  from  the  finger  key ;  the  associative  reaction  is  then,  in 
outward  form,  the  counterpart  of  the  cognitive  and  discriminative 
reactions.  It  may  also  (2)  be  the  utterance  of  the  word  which 
names  the  associated  idea:  of.  the  selective  reaction  (cognitive 
type).  Or  again  (3)  we  may  tell  O  that  he  will  be  shown  pic- 
tures which  represent  articles  either  of  dress  or  of  furniture,  and 
that  articles  of  dress  are  to  be  answered  by  the  left-hand,  and 
articles  of  furniture  by  a  right-hand  movement;  so  that  the  re- 
action movement  implies  the  subsumption  of  the  given  impression 
under  the  general  category  of  Mress '  or  of  *  furniture.'  This 
mode  of  reaction  [cf.  the  selective  reaction,  discriminative  type) 
may  be  extended  to  five  or  ten  correlations  of  movement  and 
category. 

EXPERIMENT  XXVI 

Make  out  a  plan  for  an  experiment  upon  the  associative  re- 
action, and  submit  it  to  the  Instructor.  The  experiment  has 
both  a  psychophysical  and  a  psychological  value  :  try  to  allow 
for  both,  in  the  time  at  your  disposal.  Remember  that  some 
time  must  be  allowed  for  practice.  Use  the  instruments  avail- 
able in  the  laboratory. 

Questions. — O  ( i )  Give  a  full  analysis  of  the  associative  con- 
sciousness, under  the  conditions  of  your  (psychological)  experi- 
ment.    [See  Question  (4),  p.  191  above.] 

E  (2)  Suggest  apparatus  and  method  for  the  following  prob- 
lems : 

ia)  associative  reactions  to  olfactory  stimuli ; 
[b)  a  simple  arrangement  of  the  associative  reaction,  with- 
out the  Hipp  chronoscope  and  its  accessories ; 
{c)  associative  reactions  to  auditory  stimuli  other  than  spo- 
ken words. 
E  and  O  (3)  Discuss  the  importance  of  the  associative  reac- 
tion for  systematic  psychology,  and  its  relation  to  the  compound 
reactions  of  the  foregoing  §. 

E  and  O  (4)  What  do  you  mean  by  a  'judgment,'  psycho- 
logically regarded.?     Do  the  judgment-reactions  spoken  of  above 


§  2/.    Compound  Reactions;  Association  195 

really  imply  judgment  ?     What  of  the  reactions  with  partially 
constrained  association  ? 

E  and  O  (5)     Discuss  the  following  definitions  of  association: 
(a)  Assoziation  ist  das,  wodurch  es  erst  moglich  wird,  dass 
ein  Erlebnis  von  einem  andern  reproduziert  werde. — Watt. 
{b)   L'  association  des  idees  signifie  la  liaison,  la  connexion, 
I'attraction  des  idees.     L'association,  la  liaison  des  idees 
n'est  pas  un  phenomene  de  conscience. — Claparede. 
{c)  A  union  more  or  less  complete  formed  in  and  by  the 
course  of  experience  between  the  mental  dispositions  cor- 
responding to  two  or  more  distinguishable  contents  of 
consciousness,  and  of  such  a  nature  that,  when  one  con- 
tent recurs,  the  other  content  tends  in  some  manner  or 
degree  to  recur  also. — Diet,  of  Phil,  and  Psych. 


CHAPTER   IV 

THE   PSYCHOLOGY   OF   TIME 

§  28.  The  Reproduction  of  a  Time  Interval. — The  reading  of 
Fechner's  Elemente  der  Psychophysik  suggested  to  the  Austrian 
physicist  E.  Mach  (b.  1838)  the  question  whether  Weber's  Law- 
holds  for  our  estimation  of  time,  as  it  holds  for  our  estimations 
of  sensible  intensity.  Mach  went  to  work  at  once,  and  between 
the  years  i860  and  1865  made  a  large  number  of  experiments. 
His  results  were  negative,  but — like  all  pioneer  results — were 
very  far  from  conclusive.  The  example  which  Mach  set  was 
quickly  followed  by  other  investigators,  and  experimental  studies 
of  the  time  consciousness  came  to  play  a  large  part  in  the  liter- 
ature of  experimental  psychology.  As  the  years  have  passed, 
the  questions  at  issue  have  been  differentiated  and  refined,  and 
methods  and  instruments  have  grown  correspondingly  delicate 
and  accurate.  We  can  do  no  more  here  than  repeat  one  of  the 
earlier  and  cruder  experiments, — an  experiment  described  by  the 
physiologist  K.  Vierordt  (18 18-1884)  in  1868. 

EXPEEIMENT  XXVII 

Materials. — Vierordt  lever  with  accessories.  Kymograph. 
Time  marker.  Arm  rest.  Soundless  metronome.  [The  Vierordt 
lever,  Fig.  85,  is  a  strip  of  brass,  which  turns  about  a  transverse 
axis,  and  carries  at  the  end  of  its  shorter  arm  a  flexible  writing 


Fig.  85. 
point.     The  lever  is  held  in  the  horizontal  position  by  a  spring, 


§  28.    TJie  Reprodtictio7t  of  a   Time  Interval  197 

whose  tension  is  regulated  by  a  collar  and  set  screw  adjusted 
to  the  support.  To  the  under  side  of  the  long  arm  of  the  lever 
is  fastened  a  short  brass  rod.  Beneath  the  rod  is  placed  a  sheet 
of  heavy  glass  ;  and  above  the  lever,  about  midway  between  the 
brass  rod  and  the  support,  an  iron  rod  moves  vertically  between 
guides  attached  to  the  standard  which  holds  the  entire  apparatus. 
Glass,  standard  and  iron  rod  are  not  shown  in  the  Fig.] 

Course  of  Experiment. — O  is  comfortably  seated  beside  the 
table  which  carries  the  apparatus,  his  eyes  closed,  his  arm  sup- 
ported in  the  arm  rest,  and  the  fingers  of  his  right  hand  extended 
just  above  the  end  of  the  long  arm  of  the  lever.  E^  taking  his 
time  from  the  soundless  metronome,  drops  the  iron  rod,  twice 
over,  from  a  constant  height.  O  thus  hears  two  sharp  sounds  of 
the  same  intensity,  embracing  the  standard  time  interval.  He 
is  himself  to  tap  the  lever  when  he  thinks  that  an  interval  has 
elapsed  equal  to  the  interval  bounded  by  the  given  sounds.  The 
glass  plate  serves  to  regulate  the  excursion  of  the  lever,  and  also 
to  furnish  a  third,  limiting  sound.  To  avoid  complication  by  the 
development  of  a  rhythm,  the  kymograph  is  stopped  and  a  pause 
is  made  between  experiment  and  experiment. 

E  thus  obtains  on  the  surface  of  the  kymograph  the  record 
of  normal  and  reaction  taps  and,  just  below  this,  the  time  record 
(say,  in  tenths  of  a  sec). 

Preliminaries. — The  experiments  should  extend,  by  quarter 
second  intervals,  from  0.25  to  about  5  sec.  ^S'must,  therefore, 
practice  the  production  of  these  intervals  by  help  of  the  sound- 
less metronome.  Even  after  practice,  however,  there  will  be  a 
good  deal  of  variation  in  the  normal  times. 

O  must  also  have  some  preliminary  practice  in  the  estimation 
of  the  times  employed.  Warming-up  experiments  must  be  made 
at  the  beginning  of  every  laboratory  session. 

Treatment  of  Results. — The  normal  times  are  presented 
(each  25  times  over)  in  haphazard  order.  At  the  end  of  the 
experiment,  E  may,  as  circumstances  suggest,  do  either  one  of 
two  things.  He  may  (i)  arrange  his  normal  times  in  groups, 
taking  first  the  times  up  to  0.25  sec,  next  the  times  from  0.26 
to  0.5  sec,  next  the  times  from  0.51  to  0.75  sec,  and  so  on. 
The  times  of  these  groups  are  averaged ;  the  corresponding  re- 


198  The  Psychology  of  Time 

production  times  are  averaged ;  and  the  two  averages  are  made 
the  basis  of  further  calculation.  Thus,  the  average  normal  time 
of  the  first  group  might  be  0.21  sec,  that  of  the  second,  0.36 
sec,  and  so  on.  Then  these  times,  0.21  and  0.36  sec,  are  com- 
pared with  the  averages  of  (9's  corresponding  times.  Or  (2)  he 
may  sort  out  the  normal  times  in  groups,  according  to  their  abso- 
lute value,  and  tabulate  only  those  intervals  which  were  exactly 
repeated,  say,  at  least  10  times  over,  in  the  course  of  the  ex- 
periment. 

The  normal  times,  in  the  order  lowest  to  highest,  form  the 
first  vertical  column  of  a  Table.  In  the  second  column  are 
entered  the  average  crude  errors,  plus  or  minus^  expressed  in  % 
of  the  normal  times.  In  the  third  are  entered  the  average  varia- 
ble errors,  also  expressed  in  %  of  the  normal  times.  A  fourth 
column  shows  the  percentage  of  positive  crude  errors ;  a  fifth 
gives  the  number  of  determinations  made  with  the  various  normal 
times. 

Questions. — E  (i)  What  conclusions  do  you  draw  from  the 
figures  of  the  Table  } 

O  (2)  What  is  the  introspective  basis  of  your  judgment  of 
equality .? 

E  and  (9  (3)  Criticise  the  method  employed,  and  suggest  an 
alternative. 

E  and  O  (4)  Suggest  an  apparatus  for  the  determination  of 
the  T>L  for  time  interv^als. 

E  and  O  ($)  Why  should  Weber's  Law  be  expected  to  hold 
of  the  discrimination  of  time  intervals  >  Were  Mach's  results 
entirely  negative .? 

E  and  O  (6)  What  are  the  principal  problems  of  the  psy- 
chology of  time } 


LIST  OF  MATERIALS 


The  following  list  includes  only  those  instruments  and  appliances  which  have 
been  prescribed  for  the  experiments  of  the  text.  Further  apparatus  will  be  needed, 
if  the  programme  of  work  is  to  be  completely  carried  out  ;  details  of  the  various 
instruments  available  will  be  found  in  Part  II. 

The  materials  may  be  obtained  from  the  C.  H.  Stoelting  Co.,  31-45  W.  Randolph 
Street,  Chicago,  111. 

I.  SPECIAL  APPLIANCES 


iEsthesiometer,  79,  103. 
Appunn's  lamella,  i. 
Arm-rest,  115,  196. 

Bell  signals,  167. 

Carrier  bracket,  115. 
Cartridge  weights,  80. 
Chin-rest,  2,  12. 

Colour  mixer,  Wundt's  triple,  90. 
Colour  mixers,  31,  66,  87,  89. 
Control    instrument    for   Hipp    chro no- 
scope,  167. 

Dark  box,  88. 

Envelope  weights,  33. 

Finger-key,  break,  167. 

Galton  bar,  76. 

Galton  whistle,  Edelmann's,  I2f. 

Hair  sesthesiometer,  von  Frey's,  I5f. 
Head-rest,  76,  88,  89. 
Hipp  chronoscope,  167. 

Kirschmann  photometer,  34,  66,  86,  89  f. 
Kymograph,    with    accessories,   19,  20, 
196. 


Limen  gauge,  von   Frey's,    17 

ments,  18  f. 
Lehmann's  acouttieter,  2^  f. 


attach- 


Metronome,  116. 

Metronome,   soundless,  68,  80,  87,^  115, 

196. 
Miinsterberg's  arm-movement  apparatus, 

104. 

Rumford  photometer,  with  white  card- 
board screen,  31. 

Sound  hammer,  167. 
Sound  pendulum,  87,  115. 

Time  marker,  20,  196. 

Touch  weights.  Scripture's,  14  f. 

Tuning  forks,  one  with  riding   weights, 

67. 
Tuning  forks,  with  screws,  68. 

Vierordt's  lever,  196. 

Weight -holders,  Fechner's,  with  weights, 
115- 


II.  GENERAL  APPLIANCES  AND   MATERIALS 


Ammeter,  134. 

Baize,  i,  12. 

Balance,  w4th  weights,  21,  33. 

Box,  cardboard,  10;  padded,  23. 


Candle,  standard,  31. 

Commutators,  167. 

Current  distributors  :    Nichols  rheostat, 

129  f.;  lamp  batteries,  I29ff. 
Current  tester,  i4of. 


199 


200 

Cushion,  167. 
Dry  cells,  167. 

Ear-plug,  2,  12,  23. 
Eye-shade,  23,  166, 

Felt,  I,  12,  67  f. 
Forceps,  25. 


Galvanometer,  167. 

Glass,  strip  of  plate,  30,  34 ;  small  plate 
of,  21  ;  plate  of  heavy,  197. 

Glasses,  grey  ('  smoked '),  29,  31  ;  col- 
oured, 30. 

Lamp,  4  c.p.  incandescent,  31. 

Metal,  small  plate  of,  21. 
Metre  bridge,  136. 
Metre  rod,  23. 
Motor,  electric,  91, 


List  of  Materials 


Plaster  mould,  20,  116,  167. 
Pole  tester,  127. 
Protractor,  metal,  66,  87,  89. 
Push  buttons,  167. 

Rheochord,  167. 
Rod,  iron,  197. 

Sand  box,  20,  116. 

Spectacle  frame  (opticians'  trial  frame), 

29.  31- 
Standard,  18;  with  arm  and  clamp,  12; 

with  guides  for  iron  rod,  197. 
Stop-watch,  67  f.,  80. 
Switch,  two-way,  167. 

Tubing,  rubber,  for  gas  burners,  91, 

Uprights,  wooden,  23,  115. 

Voltmeter,  134. 

Watch,  23. 

Welsbach  burners,  91. 
Wire,  167. 


Piano  hammer,  67  f. 

III.  PAPER,  CLOTH,  DRAWING  MATERIALS,  ETC. 

Observation  tube,  black,  66,  89. 


Cardboard,  strip  of,  24 ;  sheet  of  black 
or  white,  34;  squares  of  black  and 
white,  86. 

Delbceuf  disc,  88  f. 

Discs,  black  and  white,  31,  66,  89. 

Dye,  16. 

Forms,  blank,  for  method  work,  3  ff . ; 
record,  10. 

Indian  ink,  86. 

Mm.  paper  scale,  79,  103. 


Papers,  Marbe's  grey,  30,  34 ;  Hering's 
grey,  66. 

Screens,  116,  167;  black,  76;  curved 
black,  31  ;  grey,  66;  muslin  window, 
66;  white,  91. 

Tape,  23,  115. 
Tickets,  cardboard,  10. 

Wax,  168. 


IV.  LABORATORY  FURNITURE 
Chair,  swivel,  169,  174.  |  Table,  high,  3 ;  low,  33,  34, 91. 


INDEX  OF  NAMES  AND  SUBJECTS 


Accumulators,  126  f. 

Acoumeter,  Lehmann's,  23  f. 

Action,  local,  in  Voltaic  cell,  122  f. 

Adjustment  of  equivalent  R,  method  of, 
70  ff. 

^sthesiometer,  von  Frey's  hair,  15  f. 

^sthesiometric  compasses,  79,  103. 

iEsthesiometry,  illustration  from,  92  ff. ; 
experiments  in,  79  f.,  103  f. 

Affective  intensity,  quantitative  deter- 
mination of,  xxxix. 

Alechsieff,  N.,  181,  184. 

Ambiguous  associations,  192. 

Ammeter,  132  ff. 

Ampere,  definition  of,  121. 

Appunn,  A.,  i. 

Arithmetical  mean,  as  most  probable 
value  of  a  set  of  observed  values, 
39,  72,  181. 

Arm-movement  apparatus,  Munster- 
berg's,  104  f. 

Association,  in  reaction  experiment, 
192  ff. ;  free,  192  ;  ambiguous  or  par- 
tially constrained,  192 ;  constrained 
or  uni vocal,  193. 

Average  deviation,  8  f. ;  see  MV. 

Average  error,  Fechner's  method  of, 
70  ff. 

Average  variable  error,  72  ;  and  MV  ox 
PE,  73- 

Batteries,  primary,  how  formed  from 
cells,  1 24  f . ;  for  high  and  low  external 
resistances,  126;  storage,  I26f. ;  lamp, 
129  ff. 

Belief  and  probability,  53  ff. 

Bell,  single  stroke,  168. 

Bias,  error  of,  in  work  by  method  of 
limits,  61  f. 

Blank  experiments,  23. 

Bottone,  S.,  133. 

Bradley,  J.,  51. 


Brightness,  determination  of  DL  for,  by 
method  of  limits,  66  f.  [cf.  ()'^. 

Brightness,  application  of  method  of 
equal  sense-distances  to,  87  ff. 

Brightness  experiment,  Ebbinghaus',  34. 

Buccola,  G.,  163. 

C,  as  symbol  of  current  strength,  121. 

Carhart,  H.  S.,  123. 

Cattell^J.  McK.,  42,  148,  158,  165. 

Cell,  the  simple  Voltaic,  122  ff. ;  open 
circuit  and  closed  circuit,  123;  Le- 
clanche,  123;  Daniell,  123  f.;  Mei- 
dinger,  124;  Grenet  or  bichromate, 
1 24 ;  arrangements  of  cells  in  batteries, 
124  f. ;  dry,  128. 

Chance,  45. 

Choice  reactions,  188. 

Chronograph,  recording,  149. 

Chronometer,  gravity,  151,  156  ff. 

Chronoscope,  Ewald,  158,  160. 

Chronoscope,  Hipp,  description  of, 
142  ff. ;  clockwork,  142  f. ;  registering 
apparatus,  143  f. ;  electromagnetic 
mechanism,  144  f . ;  four  possible  ar- 
rangements of,  145  f. ;  three  arrange- 
ments of,  for  reaction  experiments, 
146  ff. ;  control  of,  149,  151  ff.,  172  f., 
176  ;  general  rules  for  use  of,  149  ff. ; 
current    strength    required   for,    167, 

175- 

Clang  stimulator,  Martius',  1 54  f . 

Claparede,  E.,  195. 

Cloud  experiment,  Fechner's,  30 ;  alter- 
native experiments,  30  f. 

Coefficient  of  variation,  183. 

Cognitive  reaction,  187  f . ;  see  also 
189  f.,  190  f. 

Commutator,  Pohl's,  148  f. 

Compound  reactions :  discrimination, 
cognition  and  choice,  185  ff. ;  associa- 
tion, 192  ff. 


202 


Index  of  Names  and  Subjects 


Conductivity,  124,  132. 

Constant  R,  method  of,  92  ff. 

Constant  i^-differences,  method  of, 
106  ff. 

Constrained  associations,  193. 

fContrast,  quantitative  determination  of, 
xxxix. 

Control  instruments  for  Hipp  chrono- 
scope,  151  ff. ;  their  place  in  the 
chronoscope  circuit,  152  f.,  169,  174; 
gravity  chronometer,  151,  158;  pendu- 
lum, 151  ;  hammer,  151  f.,  155  ;  direc- 
tions for  use  of,  172  f.,  176. 

Correlation  of  sense-distance  with  stimu- 
lus magnitude,  xxvii.,  xxxii.  ff .,  xxxviii., 
25  ff.,  83. 

Current,  electric,  121 ;  unit  of,  121 ;  rule 
for  determining  direction  of,  1 28 ;  dis- 
tribution of,  through  the  laboratory, 
128  £f.;  distribution  of,  in  a  simple 
circuit,  129;  measurement  of,  I32f. 

Current  tester,  140  f. 

Curves,  construction  of,  41  ;  equations 
of,  41,  43  f. ;  illustrations  of  exponen- 
tial, 42  f.  {cf.  26  f.,  29). 

Cutaneous  extents,  equation  of,  79  f. 

D,  as  symbol  of  distance  or  difference, 
92. 

Dark  box,  Delboeuf's,  88  f. 

Delbceuf,/.  R.  Z.,  xxviii,  87  ff.,  91. 

Descartes,  R.,  xxiv. 

Dessoir,  M.,  160,  165. 

Dieize,  G.,  124. 

Difference,  just  noticeable,  as  unit  of 
mental  measurement,  xxxiv.  f.,  xxxvi. 

Difference  limen,  xxxvii. ;  fourfold  de- 
termination of,  xxxviii. ;  absolute  and 
relative,  xxxviii. ;  see  DL. 

Differential,  49. 

Discriminative  reaction  186  f.  ;  see  also 
188  f.,  190. 

Discs,  Delboeuf's,  88  f.;  Wundt's  ar- 
rangement of,  for  work  by  method  of 
equal  sense-distances,  90. 

Disparity  of  sense-distance  and  stimulus 
magnitude,  xxvii.,  xxxiii.  f.,  25  ff. ; 
analogy  of  the  tangent  galvanometer, 
xxviii.  f.,  xxxvi. 

Distribution,  178  (^  48)  ;  curve  of,  179 

DL,   as    symbol    of    difference    limen. 


xxxvii. ;  fourfold  determination  of, 
xxxviii. ;  absolute  and  relative, 
xxxviii.,  62  f . ;  a  phenomenon  of 
'  friction,'  xxxii.,  xxxvi. ;  an  ideal 
value,  56,  59 ;  a  variable  value,  59 ; 
definition  of,  59  ;  of  method  of  limits, 
59  f. ;  magnitude  and  course  of,  as 
shown  by  method  of  limits,  62  f.; 
curve  of  distribution  of,  in  method  of 
constant  i^-diiferences,  107. 

Danders,  F.  C,  185  f. 

Donders'  a,  b,  c  methods  of  reaction, 
185  f. 

D.  P.,  as  symbol  of  difference  of  poten- 
tial, 120. 

Dumreicher,  O.,  160. 

Duration  of  sensation,  problems  of,  xxxi. 

Dynamos,  simple  direct  current,  136  f. ; 
alternating  current,  137  ;  excitation  of 
field  coils,  137  ;  series,  shunt  and  com- 
pound wound,  137  f . ;  conversions  of 
direct  current,  139. 

e,  numerical  constant,  44. 

E,  as  symbol  of  E.  M.  F.  or  electromo- 
tive force,  121. 

Ebbinghaus,  H.,  xxviii.,  34. 
Edelmafin,  M.  T.,  12  i. 
Electromotive  force,  definition  of,  120. 
E.  M.  F,  symbol  of  electromotive  force, 

120. 
Equal  sense-distances,  method  "of,  81  ffi 
Equal-appearing    intervals,    method   of, 

81  ff. 
Equations,  of  curves,  41,  43  f. ;  normal, 

in   method   of    constant    R,   102;    in 

method  of  constant  ^'^-differences,  112. 
Equivalent     R,     determination    of,    by 

method  of  constant  R,  104  f. 
Equivalents,  method  of,  77  ff. 
Error,  average  variable,  72  ;  and  MV ox 

Error,  Gauss'  law  of,  -i^-T^  ff.,  97 ;  curve 
of,  41  ff.;  mechanical  illustration  of 
cause  of  curve  of,  46 ;  illustration  of 
a  normal  distribution,  48. 

Error  function,  50. 

Error  function  complement,  50. 

Error  of  mean  square,  64 ;  cf.  Standard 
deviation. 

Errors,  accidental,  39,  58,  60. 

Errors,  constant,  39,  57  f.,  39  f.,  74  ff.. 


Index  of  Names  and  Subjects 


205 


79,  106,  113,  149;  determination  of,  in 
method  of  limits,  65  ;  direction  of,  in 
method  of  limits,  65  ;  determination 
of,  in  method  of  average  error,  73, 
74  f. ;  in  method  of  equivalents,  79  ; 
in  method  of  constant  ^^-differences, 
106  f.,  113. 

Errors  of  observation,  38  ff.,  59  ;  mathe- 
matical theory  of,  40 ;  distribution 
of,  40  ff.,  51  ;  probability  of,  47  £f. ;  in 
Fechner's  method  of  average  error, 
70  ff. 

Errors,  systematic,  39 ;  see  Errors,  con- 
stant. 

Errors,  variable,  58,  60,  61  ;  of  chrono- 
scope,  149,  172  f.,  176;  see  Error, 
average  variable. 

EwaldJ.  R.,  158. 

Exercises,  practical,  in  electrical  meas- 
urement, 141  ;  on  the  compound  re- 
action, 191. 

Expectation  and  probability,  53  ff, ;  de- 
pendence of,  on  mental  constitution, 
54  ;  limits  of,  55  ;  error  of,  in  method 
of  limits,  57  f.,  61. 

Experiment,  Ebbinghaus'  brightness,  34  ; 
Fechner's  cloud,  30;  Fechner's 
shadow,  32  ;  Sanford's  weight,  'i>'}>  ^- ! 
Vierordt's,  on  reproduction  of  a  time 
interval,  196  ff. 

Experiments,  blank,  23. 

Experiments,  preliminary,  value  of,  2  f,, 
I3f.,  67  ;  disposition  of,  7  ;  directions 
for,  2of.,  68,  77,80,87,  103, 115,  ii6f., 
176,  194,  197;  see  Practice,  Warming 
up. 

Exposure  apparatus,  Roemer's,  193. 

Extent  of  sensation,  problems  of,   xxxi. 

Extents,  visual,  equation  of,  76  f. ;  cu- 
taneous, equation  of,  79  f. 

Fatigue,  in  serial  work,  4, 6  ;  of  skin,  20, 
103 ;  limits  of,  55 ;  in  method  of 
limits,  57  f.,  60;  in  method  of  average 
error,  74  ;  in  method  of  equivalents, 
78,  80;  in  reaction  experiments,  176. 

Fechner,  G.  T.,  xxiii.f.,  xxvi.f.,  xxxiv.  f., 
xxxvi.,  xxxviii.,  28  f.,  30,  3^2,  42,  65, 
70,75»  77,80,  99,  102,  III,  114,  1151, 
196. 

Fechner's  fundamental  table,  99,  iii, 
114. 


Finger  keys,  164  f.;  five-finger  key,  189. 

Formulae,  of  measurement  in  general, 
xix.,  xl. ;  for  law  of  affective  inten- 
sity, xxxix. ;  for  law  of  memory,  xl. ; 
for  MV,  8,  182  ;  for  Weber's  Law,, 
28  ;  for  Kirschmann's  photometer,  37  ;. 
for  error  curve,  44 ;  for  probability  of 
errors,  47,  49  ff. ;  for  FEy  52  f.,  64  f.,. 
76 ;  for  error  of  mean  square,  64 ;. 
for  constant  errors,  65,  75,  113;  for 
equivalence,  79 ;  for  sound  pendulum,. 
81  ;  for  ^2  in  method  of  equal  sense- 
distances,  86  (cf.  96) ;  for  RL,  iiv 
method  of  constant  R,  96,  97  f.  ;  for 
DL,  in  method  of  constant  i?-differ- 
ences,  no;  for  interrelation  of  elec- 
trical units,  121  f.,  125;  for  arithmeti- 
cal mean,  in  reaction  experiments, 
181 ;  for  relative  variability,  182. 

Fractionation  of  results,  in  method  of 
average  error,  74. 

Free  associations,  192. 

Frequency  polygon,  179. 

voii  Frey^  M.,  15,  17. 

Gallon,  F.,  12,  46,  48,  76. 

Galton  bar,  76. 

Galvanometer,  133,  149,  167  f . ;  analogy 

of  the  tangent,  xxviii.  ff.,  xxxvi. 
Gauss,  K.  F.,  43,  98,  104. 
Gauss'  Law,  see  Error,  Gauss'  Law  of. 
Glaisher,  J.  W.  L.,  50. 
Glazebrook,  R.  T.,  137. 
Grey  papers,  Marbe's,  30,  34  ;  Hering's, 

66. 

h,  as  measure  of  precision,  44  f.,  rela- 
tion of,  to  FE,  52;  determinations  of, 
102,  112. 

Habituation  error,  in  method  of  limits, 
58,61. 

Hair  aesthesiometer,  15  f. 

Hairs,  as  cutaneous  stimulators,  16,  157. 

Hammer,  control,   151    f.,   155;    sound, 

154- 
Ilerbart,/.  F,  xxiv.,  xxxvii. 
Hering,  E.,66,  115. 
Hipp,  M.,141  f.,  167,  178,  194, 
ITq^er,  ^.,  26  f. 

I,  as  symbol  of  current  strength,  121. 
Instructions    to     O,   in    serial    method 
work,  6, 


204 


Index  of  Names  and  Subjects 


Intensity  of  sensation,  problems  of, 
XXX. ;  Fechner's  interest  in,  xxxvi.  f. 

Interruptor,  Krille's,  157. 

Introspection,  xxii.,  xxv.  £.,  xxvii., 
xxxiv.,  141  f . ;  of  lowest  tones,  9; 
questions  involving,  xlL,  10,  14,  22,  25, 
32,  37.  69,  77,  81,  92,  104,  106,  118  f., 
183  f.,  191  f.,  194  f.,  198- 

Inversion,  of  the  first  order,  94  f.,  98, 
108  f. ;  of  the  second  order,  95,  108  f. 

J.  n.  d.,  as  unit  of  mental  measurement, 
xxxiv.  f.,  xxxvi.,  25. 

J.  n.  d.,  method  of,  55  ff. 

/astrow,/.,  159,  164,  166. 

Judgment,  direction  of,  in  work  by 
method  of  limits,  61  ;  standard  of,  in 
method  work,  2 ;  reaction  upon,  193. 

Judgments,  nature  and  direction  of,  in 
method  of  constant  ^^-differences, 
106  ;  distribution  of,  in  same  method, 
109. 

Kant,  I.,  xxiv. 

Keys,  reaction,   164  ff.  ;  telegraph  key, 

164;     Jastrow's      finger     key,     164; 

Scripture-Dessoir     finger     key,     165 ; 

Cattell's     lip     key,     165;     Jastrow's 

speech   key,    165   f . ;    Cattell's   voice 

key,  147  f.,  158,  166. 
Kiesow,  F.,  161. 

Kirschmann,  A.,  34  £f.,  66  f.,  86,  89,  91. 
Krille,  C,  157. 

Lamella,  Appunn's,  i. 
Lamp  batteries,  129  ff. 
Laftge,  Z.,  156. 
Laplace,  P.  S.  de,  51. 
Least  differences,  method  of,  55  ff. 
Lehmann,  A.,  23. 
Lever,  Vierordt's,  196. 
Lifting  weights,  two  methods  of,  T17. 
Limen,  definition  of,  xxxvii. 
Limen  gauge,  17  ;  apparatus  for  regulat- 
ing, 18  f. 
Limits,  method  of,  55  ff. 
Limits  of  sensation,  xxx.  ff. 
Lip  key,  Cattell's,  165. 

Mach,  E.,  196,  198. 
Marbe,  K.,  30,  32,  34. 
Martin,  L. /.^  115. 


Martius,  G.,  155. 

Masson,  V.,  139. 

Mean  gradations,  method  of,  81  ff. 

Mean  variation,  8  f . ;  see  MV. 

Measurement,  object  and  result  of  physi- 
cal, xix.  ;  units  of,  conventional,  xx., 
but  accurate,  xxii.,  and  interrelated, 
xxiii, ;  typified  by  linear  measurement 
in  space,  xx.  ;  always  involves  three 
terms,  xx.,  xxxii. ;  mental,  xxi.  f., 
xxxii.  ff.  ;  see  Mental   measurement. 

Median,  182. 

Memory,  quantitative  investigation  of, 
xl. 

Mental  measurement,  xxi.  ff . ;  no  natural 
unit  of,  xxii. ;  no  interrelation  of  units 
of,  xxiii. ;  difficulties  of,  intrinsic  and 
extrinsic,  xxiii.  f. ;  Fechner's  work 
upon,  xxiii.  f.,  xxvi. ;  material  of,  not 
sensation  but  sense- distance,  xxiv.  ff. ; 
schema  of,  xxxii.  ff.  ;  just  noticeable 
difference  as  unit  of,  xxxiv.  f .,  xxxvi.  ; 
fundamental  problems  of,  xxxviii.  {cf. 
xxix.  ff.). 

Meri'iman,  M.,  64. 

Method  of  determination  of  qualitative 
RL  for  tones,  3  ff.,  10 ;  of  determina- 
tion of  qual.  TR  for  tones,  14 ;  of 
determination  of  intensive  RL  for 
pressure,  15,  17,  21  ;  of  determination 
of  intensive  RL  for  sound,  23,  25  ;  of 
Fechner's  cloud  experiment,  30 ;  of 
Fechner's  shadow  experiment,  32  ;  of 
the  '  psychophysical  series,'  2)Z  ^'» 
34  f. ;  of  photometry  (Kirschmann's 
photometer),  36  f. ;  of  least  squares, 
100,  102  ;  of  reaction  experiment, 
170  ff.,  175  f.,  185  f. ;  of  Vierordt's 
experiment,  197  ;  see  Metric  methods. 

Metre  bridge,  136. 

Metric  methods,  xxiii.,  xxxii.  f.,  xxxvi., 
xxxviii. ;  method  of  limits  (just  no- 
ticeable differences,  least  differences, 
minimal  changes),  55  ff. ;  illustration 
from  brightness,  55  ff.;  sources  of 
error  in,  62  ;  test-values  of,  62  ff. ; 
method  of  average  error  (reproduc- 
tion, adjustment  of  equivalent  R), 
70  ff. ;  illustration  from  visual  extents, 
70  ff. ;  test-values  of,  72  ff. ;  method 
of  equivalents,  77  ff. ;  illustration  from 
aesthesiometry,     78   f. ;     formula    for 


Index  of  Names  and  Subjects 


205 


equivalence,  79 ;  method  of  equal 
sense-distances  (mean  gradations, 
supraliminal  differences,  equal-appear- 
ing intervals),  81  ff.;  illustration  from 
sounds,  81  ff.  ;  formula  for  calculation 
of  ^2,  86  {cf.  96) ;  method  of  constant 
R  (right  and  wrong  cases),  92  ff. ; 
illustration  from  assthesiometry,  92  f. ; 
inspection  of  results,  93  ff, ;  calcula- 
tion of  the  RL,  first  procedure,  95  ff. ; 
formula,  96  ;  second  procedure,  97  ff . ; 
formula,  98  ;  application  of  the  method 
to  the  determination  of  equivalent  R^ 
104  f. ;  method  of  constant  i'^-differ- 
ences  (right  and  wrong  cases),  106  ff. ; 
illustration  of  results  with  lifted 
weights,  107 ;  inspection  of  results, 
107  ff. ;  calculation  of  the  DL,  first 
procedure,  109  f.,  112;  second  proce- 
dure, no  ff.,  112. 

Metric  system,  xx. 

Minimal  changes,  method  of,  55  ff. 

Mode,  182. 

Modulus,  =  -7—,  ^x. 

h     ^^     • 

Moldenhauer,  W.,  162,  166. 

Motors,  direct   current,    138  f.  ;  uses  of, 

in     psychological      laboratory,     139; 

alternating  current  or  induction,  139; 

hints  for  care  of,  140. 
Mailer,  G.  E.,  77,  loi  f.,  in,  115. 
Miiller's  table  of  coefficients  of  weights, 

lOI,   III. 

Multiple-series,  connection  of  cells  in, 
126. 

MV,  as  measure  of  variability,  8  f.,  11, 
44,  64,  73,  78  f.,  83,  86,  172  f.,  176, 
182  ;  and  FE,  65  ;  and  average  varia- 
ble error,  73. 

Nichols,  E.  L.,  128  ff.,  131,  139,  141. 

Ohm,  G.  S.,  121. 
Ohm,  definition  of,  120  f. 
Ohm's  Law,  121,  125,  132,  135. 
Optical  illusions,  quantitative  determina- 
tion of,  xl. ;  pendulum,  156,  183. 

/,  as  symbol  of  time  error,  79,  113. 

IT,  numerical  constant,  44. 

Parallel,  connection  of  cells  in,  125  f. 


FE,  as  measure  of  variability,  52,  64, 
75  f .  ;  and  modulus,  53;  of  single 
observation  and  of  mean,  64,  183; 
and  error  of  mean  square,  64 ;  and 
MV,  65,  76 ;  and  average  variable 
error,  'jt,;  of  standard  deviation  or 
error  of  mean  square,  183. 

Pendulum,  control,  151  ;  optical,  156, 183. 

Personal  equation,  xxxvii. 

Photometer,  Rumford's,  31 ;  Kirsch- 
mann's,  34  ff.,  66,  86 ;  formula  for 
Kirschmann's,  37. 

Photometric  units,  67. 

Polarisation,  in  Voltaic  cell,  122  f. 

Pole  tester,  127. 

Fohl,  G.  F.,  147  f.,  167. 

Potential,  definition  of  electrical,  120; 
difference  of,  1 20  ff . 

Practice,  in  serial  work,  5  ;  limits  of,  55  ; 
in  method  of  limits,  57  ;  in  method 
of  equivalents,  80;  distribution  of,  in 
method  of  equal  sense  distances,  84  ; 
in  reaction  experiments,  176  f . ;  see 
Experiments,  preliminary,  Warming 
up. 

Preliminary  experiments,  see  Experi- 
ments, preliminary. 

Pressure,  determination  of  RL  for,  by 
touch-weights,  14  f.;  by  hair  aesthesi- 
ometer,  1 5  ff .  ;  by  Hmen  gauge,  1 7  ff . 

Pressure,  electrical,  120;  unit  of,  120; 
distribution  of,  in  simple  circuit,  1 29 ; 
measurement  of,  132  ff. 

Principal  error,  in  method  of  average 
error,  75  ;   in  method  of   equivalents, 

79. 

Probability,  45  ff.,  59;  and  relative  fre- 
quency, 47  ff.,  93  f.,  108;  formulas  of, 
49  ff. ;  and  expectation  or  belief,  53  ff. 

Probability  integral,  50  f.,  52  f. 

Probable  error,  52  ;  see  FE. 

Psychology,  see  Mental  measurement, 
Quantitative  psychology. 

Psychophysics,  xxiii.,  xxxviii. 

q,  as  symbol  of  space  error,  75. 
QL,  as  symbol  of  quotient  limen,  64. 
Quality     of     sensation,     problems     of, 

XXX.  f. 

Quantitative  psychology,  Fechner's 
service  to,  xxiii. ;  attitude  of  Kant 
and    Herbart    to,    xxiv. ;    object    of 


2o6 


Index  of  Names  and  Subjects 


measurement  in,  xxiv.  ff.,  xxxii,  f. ; 
problems  of,  xxviii.,  xxix.  ff.,  xxxvi., 
xxxviii.  ff . ;  growth  and  status  of, 
xxxvi.  f. ;  Weber's  Law  as  gateway 
to,  xxxviii. ;  coextensive  wnth  quali- 
tative psychology,  xxxviii.  f. ;  illus- 
trated by  reference  to  brightness  con- 
trast, xxxix.  f. ;  to  affective  intensity, 
xxxix.  ;  to  optical  illusions,  xl. ;  to 
memory,  xl. 

Questions,  on  measurement  in  general, 
xl.  f. ;  on  RL  for  tones,  9f. ;  on  TR 
for  tones,  14;  on  RL  for  pressure, 
22 ;  on  RL  for  sound,  25  ;  on  Fech- 
ner's  cloud  and  shadow  experiments, 
32  f. ;  on  Sanford's  weight  and  Eb- 
binghaus'  brightness  experiments,  37  ; 
on  method  of  limits,  69  ;  on  method  of 
average  error,  77 ;  on  method  of 
equivalents,  80  f . ;  on  method  of  equal 
sense-distances,  91  f . ;  on  method  of 
constant  R,  103  f.;  on  determination 
of  equivalent  R  by  method  of  constant 
R,  106;  on  method  of  constant  R- 
differences,  1 18  f. ;  on  electrical  meas- 
urements, 141  ;  on  technique  of  simple 
reaction,  1C6  f . ;  on  types  of  simple 
reaction,  183  ff. ;  on  compound  re- 
actions (discrimination,  etc.),  191  f . ; 
on  associative  reaction,  194  f. ;  on 
Vierordt's  experiment  and  the  psy- 
chology of  time,  198. 

Quotient  limen,  63  f. 

R,  r,  as  symbol  of  stimulus,  xxxvii. 

R,  as  symbol  of  electrical  resistance, 
121. 

Range  of  results,  in  reaction  experi- 
ments, 182. 

Reaction,  technique  of  simple,  141  ff. ; 
three  arrangements  of  chronoscope 
circuit,  146  ff .  ;  experiment  on  three 
types  of  simple,  167  ff. ;  disposition  of 
apparatus  for  simple,  169  f.,  173  ff., 
183;  course  of  experiment,  170  ff. ; 
results,  statistical  treatment  of,  177  ff. ; 
compound,  185  ff.,  192  ff . ;  table  of 
procedures,  185  f . ;  discriminative, 
186  f.;  cognitive,  187  f . ;  choice,  188; 
selective  with  discrimination,  188  f. ; 
with  cognition,  189  f. ;  volitional  w'ith 
discrimination,    190;    with   cognition. 


i9of. ;   associative,  types  of,   192  ff.; 

technique  of  193  f. 
Reaction  times,  xxxvii.,  141  f.,  177. 
Reiz,  xxxvii. 
Reiz/idhe,  xxxvii. 
Relative  variability,  182. 
Reproduction,  method  of,  70  ff.,  197  f. 
^-error,    nature   of,   xxvi.  ;  in    Fechner, 

xxvi.  f. 
Resistance,    electrical,    1 20  f . ;    rule  for 

determining  joint,  132;    measurement 

of,  134  ff. 
Rheochord,  168. 
Rheostat,  Nichols',  129  f.,  131  f. 
Right  and  wrong  cases,  method  of,  92  ff., 

106  ff. 
RL,  as  symbol  of  stimulus  limen,  xxxvii. ; . 

fourfold    determination    of,    xxxviii. ; 

a   phenomenon    of    'friction,'  xxxii., 

xxxvi. ;  qualitative,  for  tones,  i  ff.,  7, 

10  ff.;  intensive,  for  pressure,   14  ff. ; 

fourfold  dependency  of,  21  ;  intensive, 

for  sound,   22  ff. ;  distribution  of,  in 

method  of  constant  A',  94. 
Rcemer  E.,  193. 
Rum  ford,  31. 
rv,  as  symbol  of  relative  variability,  182. 

,5*,  s,  as  symbol  of  sensation,  xxxvii. 

J,  as  symbol  of  principal  error,  75,  79. 

a,  as  symbol  of  yoVir  ^^^'i  ^42« 

Sanford,  E.  C,  23- 

Schwelle,  xxxvii. 

Scripture,  E.  W .,  14,  I29f.,  159,  165. 

Selective  reaction  with  discrimination, 
188  f . ;  with  cognition,  J89  f. 

Sensation,  not  measurable,  xxv.,  xxvii. ; 
stronger  does  not  contain  weaker, 
xxv.  ff. ;  correlation  of  attributes  of, 
with  properties  of  stimulus,  xxix.  ff. ; 
a  continuous  function  of  stimulus, 
xxxvi.  ;  see  Duration,  Extent,  Inten- 
sity, Quality. 

Sense-distance,  the  material  of  measure- 
ment of  '  sensation,'  xxv.,  xxxii.,  25  ff. ; 
see  Correlation,  Disparity. 

Sensibilometer,  Dessoir's,  160. 

Series,  connection  of  cells  in,  125  f. 

Series,  construction  of,  in  method  work, 
4  ;  starting-point  of,  4,  60 ;  order  of, 
5,  61  ;  length  of  successive,  61. 

Shadow  experiment,  Fechner's,  32. 


hidex  of  Names  and  Subjects 


207 


Shutter,  Jastrow's,  159. 

Simple  reaction,  technique  of,  141  ff . ; 
experiments  on,  167  ff. 

Smell  stimulator,  Moldenhauer's,  161  f  ; 
Buccola's,  162  f. 

Smith,  W.  G.,  184. 

Sound  hammer,  154. 

Sound  pendulum,  81  f.,  87  ;  formula  for 
intensity  of  sound,  81. 

Sound  stimulator,  153  f. 

Space,  measurement  of,  typical  of  all 
measurement,  xx. 

Space  error,  in  method  of  limits,  57  f. ; 
60,  65  f. ;  positive  and  negative,  65, 
75 ;  in  method  of  equal  sense-dis- 
tances, 90  f . 

Standard  deviation,  182;  see  Error  of 
mean  square. 

Steps,  size  of,  in  serial  method,  4, 
60. 

Stimulators,     in     reaction     experiment, 

153  ff . ;  noise  stimulator  (dropped 
ball),  153  f . ;  sound  hammer,  154; 
telephone  arrangement  for  tone  or 
noise,  1 54  ;  clang  stimulator,  Martius', 

154  f.  ;  light  pendulum,  156;  gravity 
chronometer,  I56ff. ;  shutter,  157,  159; 
pressure  (Scripture's  touch-key,  Des- 
soir's  sensibilometer),  157,  159  f . ; 
electrical,  158  ff . ;  temperature  (von 
Vintschgau's  thermophor,  Kiesow's 
cone),  159  ff . ;  olfactory  (Molden- 
hauer's, Buccola's,  Zwaardemaker's), 
161  ff. ;  gustatory,   163. 

Stimulus  limen,  xxxvii. ;  fourfold  deter- 
mination of,  xxxviii. ;  see  RL. 

Storage  batteries,  126  f. 

Sum,  limit  of,  49  f. 

Supraliminal  differences,  method  of, 
81  ff. 

Surface  of  frequency,  179. 

Switch,  two-w^ay,  168. 

tz=.hx,  50  ff. ;  meaning  of  the  symbol, 
53  ;  values  of,  in  Fechner's  fundamen- 
tal table,  99 ;  must  be  weighted,  100. 

Table,  Fechner's  fundamental,  99,  iii, 
114;  Miiller's,  of  coefficients  of 
weights,  loi.  III;  of  frequencies, 
178. 

Taste  stimulators,  163. 

Technical  terms,  xxxvii.  f. 


Telephone  snapper,  as  current  tester, 
141  ;  stimulator,  for  tone  or  noise,  154. 

Telescope,  reading,  188. 

Temperature  stimulator,  Kiesow's,  161  ; 
see  Thermophor. 

Terminal  stimulus,  xxxvii. ;  fourfold  de- 
termination of,  xxxviii. ;  see  TR. 

Terms,  technical,  xxxvii.  f. 

Thermophor,  von  Vintschgau's,  i6i. 

Thompson,  S.  R.,  135!,  138. 

Thorndtke,  E.  L.,  178. 

Time  error,  in  method  of  equivalents, 
79 ;  in  method  of  constant  >V-differ- 
ences,  106,  113. 

Time  interval,  reproduction  of  a,  197. 

Tone,  lowest  audible,  i  ff.,  10  ff . ;  high- 
est audible,  12  ff. ;  determination  of 
DL  by  method  of  limits,  67  ff. 

Tones,  deep,  intrinsically  weak,  i  ;  in- 
trospection of,  9. 

Touch- key.  Scripture's,  159. 

Touch-weights,  Scripture's,  14  f. 

TR,  as  symbol  of  terminal  stimulus, 
xxxvii. ;  fourfold  determination  of, 
xxxviii. ;  qualitative,  for  tone.s,  12. 

Tuning  forks,  67  f.,  154. 

Unit  of  mental  measurement,  xxxiv.  £., 
xxxvi. ;  of  measurement  a  distance, 
not  a  point,  47,  178  ;  electrical  units, 
i2of. ;  see  Metric  system. 

Upper  limit  of  sensation,  xxxi.  f. 

Variability,  measures  of,  see  Average 
variable  error,  Error  of  mean  square,  h^ 
MV,  RE. 

Variability,  relative,  182. 

Variation,  coefficient  of,  183. 

Vierordt,  K.,  196. 

von  Vintschgau,  M.,  161. 

Visual  extents,  equation  of,  76  f. 

Voice  key,  Cattell's,  147  f.,  158,  166. 

Volitional  reaction  with  discrimination, 
190;  with  cognition,  190  f. 

Volt,  definition  of,  120. 

Voltmeter,  132  ff. 

Warming  up,  67,  68,  77,  80,  87,  90,  103, 
115,  118,  176  f.,  197;  see  Experiments, 
preliminary.  Practice. 

Wait,  H.J.,  195. 


208 


Index  of  Names  and  Subjects 


Weber^  E.  H.,  xxxviii.,  25  ff.,  28  f.,  42, 
113  f.,  115,  195,  198. 

Weber's  Law,  xxxviii.,  25  ff. :  Fechner's 
formulation  of,  28 ;  demonstrations 
of,  29  ff. ;  curves  showing  relation  of 
S\.o  R  in,  26  f.,  29,  42;  test  of,  by 
method  of  constant  j??-differences, 
113  f. ;  for  time  intervals,  196. 

Weight  experiment,  Sanford's,  t^i  f. ;  in 
method  of  constant  A'-differences,  107, 
115  ff. ;  materials  for  Fechner's  115  f. 

Weights,    envelope,    -^^-y  cartridge,  80; 


Fechner's,     115   f . ;    Fechner's,     two 

methods  of  lifting,  117. 
Wheatstone,  C,  135. 
Wheatstone  bridge,  135  f. 
Witmer^  Z.,  167. 
Wright,  A.,  129  f. 
Wundt,    W.   M.,   xxxvii.,    29,    124,    156, 

184. 
Wundt's  ^-method  of  reaction,  185  f. 

Yerkes,  R.  M.,  179,  184. 

Zwaardemaker^  H.,  163. 


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